Abstract

The existence of quasi-Bragg conditions (QBCs) in Thue–Morse dielectric multilayers (TMDMs), based on high reflectance and large forbidden gap regions in its transmission spectra, is presented. In contrast to a single traditional Bragg condition (TBC) in periodic bilayers, there are three QBCs in TMDMs. The formula for each of these QBCs is derived. The results show that one of these QBCs just overlaps the TBC. However, a singular point with perfect transmission exists in this QBC at the quarter-wave stack, at which high reflection exists in the periodic bilayers.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (2)

2008 (3)

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008).
[CrossRef]

2007 (2)

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

B. S. Williams, Nat. Photonics 1, 517 (2007).
[CrossRef]

2006 (3)

I. Nusinsky and A. A. Hardy, Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Y. Taniyasu, M. Kasu, and T. Makimoto, Nature 441, 325 (2006).
[CrossRef]

L. Moretti, I. Rea, L. Rotiroti, I. Rendina, G. Abbate, A. Marino, and L. De Stefano, Opt. Express 14, 6264 (2006).
[CrossRef]

2005 (1)

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

2000 (1)

X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B 62, 14020 (2000).
[CrossRef]

1998 (2)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

E. Yablonovitch, Opt. Lett. 23, 1648 (1998).
[CrossRef]

1997 (2)

N. H. Liu, Phys. Rev. B 55, 3543 (1997).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

1995 (1)

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

1994 (1)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

1992 (1)

C. S. Ryu, G. Y. Oh, and M. H. Lee, Phys. Rev. B 46, 5162 (1992).
[CrossRef]

1984 (2)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

1958 (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Abbate, G.

Anderson, P. W.

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Bechinger, C.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

Blech, I.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Capasso, F.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Chaikin, P. M.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Chang, C. H.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Chen, C. H.

W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Chen, C. T.

W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Chen, X.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

Cheng, Y. H.

Cho, A. Y.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

De Stefano, L.

Dubey, G. S.

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

Faist, J.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Feng, S.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Grimm, U.

X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B 62, 14020 (2000).
[CrossRef]

Gumbs, G.

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

Hardy, A. A.

I. Nusinsky and A. A. Hardy, Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Helden, L.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

Hsueh, W. J.

Huang, D.

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

Huang, K. C.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Hutchinson, A. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Jeong, H.-C.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Jiang, X.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Joannopoulos, J. D.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

John, S.

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

Kasu, M.

Y. Taniyasu, M. Kasu, and T. Makimoto, Nature 441, 325 (2006).
[CrossRef]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Lee, M. H.

C. S. Ryu, G. Y. Oh, and M. H. Lee, Phys. Rev. B 46, 5162 (1992).
[CrossRef]

Li, M.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

Lin, Z. J.

Liu, N. H.

N. H. Liu, Phys. Rev. B 55, 3543 (1997).
[CrossRef]

Lu, W.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

Mahmoud, B. S.

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

Makimoto, T.

Y. Taniyasu, M. Kasu, and T. Makimoto, Nature 441, 325 (2006).
[CrossRef]

Marino, A.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Mikhael, J.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

Moretti, L.

Nusinsky, I.

I. Nusinsky and A. A. Hardy, Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Oh, G. Y.

C. S. Ryu, G. Y. Oh, and M. H. Lee, Phys. Rev. B 46, 5162 (1992).
[CrossRef]

Rea, I.

Rechtsman, M. C.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Rendina, I.

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Roth, J.

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

Rotiroti, L.

Ryu, C. S.

C. S. Ryu, G. Y. Oh, and M. H. Lee, Phys. Rev. B 46, 5162 (1992).
[CrossRef]

Salman, A.

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

Schreiber, M.

X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B 62, 14020 (2000).
[CrossRef]

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Sirtori, C.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Sivco, D. L.

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Steinhardt, P. J.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Taniyasu, Y.

Y. Taniyasu, M. Kasu, and T. Makimoto, Nature 441, 325 (2006).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Torquato, S.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Wang, H.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

Wang, S. W.

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

Wang, X.

X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B 62, 14020 (2000).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

Williams, B. S.

B. S. Williams, Nat. Photonics 1, 517 (2007).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Wun, S. J.

Yablonovitch, E.

Yariv, A.

A. Yariv and P. Yeh, Optics Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1988).

Yeh, P.

A. Yariv and P. Yeh, Optics Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1988).

Yi, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Zhang, Y.

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

S. W. Wang, X. Chen, W. Lu, M. Li, and H. Wang, Appl. Phys. Lett. 90, 211113 (2007).
[CrossRef]

X. Jiang, Y. Zhang, S. Feng, K. C. Huang, Y. Yi, and J. D. Joannopoulos, Appl. Phys. Lett. 86, 201110 (2005).
[CrossRef]

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

Nat. Photonics (1)

B. S. Williams, Nat. Photonics 1, 517 (2007).
[CrossRef]

Nature (3)

Y. Taniyasu, M. Kasu, and T. Makimoto, Nature 441, 325 (2006).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature 390, 671 (1997).
[CrossRef]

J. Mikhael, J. Roth, L. Helden, and C. Bechinger, Nature 454, 501 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[CrossRef]

Phys. Rev. A (1)

W. J. Hsueh, C. T. Chen, and C. H. Chen, Phys. Rev. A 78, 013836 (2008).
[CrossRef]

Phys. Rev. B (5)

C. S. Ryu, G. Y. Oh, and M. H. Lee, Phys. Rev. B 46, 5162 (1992).
[CrossRef]

N. H. Liu, Phys. Rev. B 55, 3543 (1997).
[CrossRef]

X. Wang, U. Grimm, and M. Schreiber, Phys. Rev. B 62, 14020 (2000).
[CrossRef]

G. Gumbs, G. S. Dubey, A. Salman, B. S. Mahmoud, and D. Huang, Phys. Rev. B 52, 210 (1995).
[CrossRef]

I. Nusinsky and A. A. Hardy, Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Phys. Rev. Lett. (3)

D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

S. John, Phys. Rev. Lett. 53, 2169 (1984).
[CrossRef]

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef]

Science (2)

J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, Science 264, 553 (1994).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef]

Other (1)

A. Yariv and P. Yeh, Optics Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, 1988).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Transmission spectra in the TMDM for generation orders v = 4 (green dots), 5 (red dashes) and 6 (blue solid). The gray areas correspond to the region of the major gap in the band structure of the TMDM for v = 10 . The parameters of the system are n A = 2 , n B = 3 , and d A = d B = 0.5 μm . (b) The frequency range of the major gap near A in Fig. 1(a) for the TMDM with generation orders v = 3 10 . The red-dashed line and the two green-dotted lines correspond to the frequencies of the center, top, and bottom of the major gaps in the band structure, respectively. The blue solid line corresponds to the frequency of the minimum value of the transmittance in the transmission spectra. The normalized frequency Ω is defined as Ω = ω D / ( 2 π c ) , where D = d A + d B .

Fig. 2.
Fig. 2.

Gap map for normal incidence in the S7 TMDM for n A = 2 , n B = 3 . The gray areas correspond to the major gaps. The blue-dashed curves and the green-dotted curves correspond to the half-wave lines, L A , m and L B , n , respectively. The black crosses, blue triangles, brown inverted triangles, and green circles, respectively, mark the centers of the left, upper, down, and right major gaps. The orange squares mark the zero gaps. The red solid lines indicate the MMG holes that correspond to the QBCs.

Fig. 3.
Fig. 3.

(a) Transmission spectra for normal incidence in the S 7 TMDM (green solid line) and in the PBDM (blue-dashed line) for P 12 . The blue triangle, the orange square, and the brown inverted triangle, respectively, correspond to the signs, A , B , and C in Fig. 2. B marks the low transmission condition in the PBDM. The gray areas correspond to the region of the major gaps in the TMDM. The blue L marks the gapwidth in the PBDM. (b) The transmission spectra and the gapwidth in the MQBC and TBC for F = 0 1 . The crosses and green circles, respectively, correspond to the signs, D and E in Fig. 2. The dark blue solid line and the light green-dashed line correspond to the TMDM and the light blue-dotted and dashed line and the dark green-dotted line correspond to the PBDM.

Fig. 4.
Fig. 4.

| E | 2 distribution in the S 7 TMDM for (a)  F = 0.8182 , Ω = 0.2308 and (b)  F = 0.6 , Ω = 0.2083 , which, respectively, correspond to the signs, E and B in Fig. 2, and in the P 12 PBDM for (c)  F = 0.6 , Ω = 0.2083 , and (d)  F = 0.6 , Ω = 0.4167 , which, respectively, correspond to the quarter-wave and half-wave stacks. The yellow and green areas symbolize materials A and B , respectively. The white areas symbolize the bonding media. The horizontal lines, I I , I R , and I T , indicate the incident, reflected, and transmitted waves, respectively. The parameters are the same as those used in Fig. 2.

Equations (6)

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

F = p σ q + p σ ,
Ω = p σ + q 2 n B cos θ B ,
F m , n ( U ) = { 1 + [ 3 n + 2 ( 1 ( 2 ) v ) ] n A cos θ A [ 3 m + 2 ( 1 ( 2 ) v ) ] n B cos θ B } 1 ,
Ω m , n ( U ) = 1 6 { 3 m + 2 [ 1 ( 2 ) v ] n A cos θ A + 3 n + 2 [ 1 ( 2 ) v ] n B cos θ B } .
d A n A cos θ A + d B n B cos θ B = { { m + n + 4 [ 1 ( 2 ) ν ] / 3 } λ 0 / 2 for U Q B C s ( m + n + 1 ) λ 0 / 2 for M Q B C s { m + n + 2 [ 1 ( 2 ) 1 ν ] / 3 } λ 0 / 2 for D Q B C s .
d A n A cos θ A + d B n B cos θ B = { ( m + n + 4 / 3 ) λ 0 / 2 for U Q B C s ( m + n + 1 ) λ 0 / 2 for M Q B C s ( m + n + 2 / 3 ) λ 0 / 2 for D Q B C s .

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