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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 11 — May. 30, 2005
  • pp: 3913–3920
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Comparing omnidirectional reflection from periodic and quasiperiodic one-dimensional photonic crystals

A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe  »View Author Affiliations


Optics Express, Vol. 13, Issue 11, pp. 3913-3920 (2005)
http://dx.doi.org/10.1364/OPEX.13.003913


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Abstract

We determine the range of thicknesses and refractive indices for which omnidirectional reflection from quasiperiodic dielectric multilayers occurs. By resorting to the notion of area under the transmittance curve, we assess in a systematic way the performance of the different Fibonacci multilayers.

© 2005 Optical Society of America

1. Introduction

Photonic crystals are periodically structured dielectric media possessing photonic band gaps: ranges of frequency in which strong reflection occurs for all angles of incidence and all polarizations. They then behave as omnidirectional reflectors, free of dissipative losses. Since the initial predictions of Yablonovitch [1

1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–62 (1987). [CrossRef] [PubMed]

] and John [2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–9 (1987). [CrossRef] [PubMed]

], photonic crystals have been attracting a lot of attention and a wide variety of applications have been suggested [3

3. A complete and up-to-date bibliography on the subject can be found at http://home.earthlink.net/~jpdowling/pbgbib.html

].

In the one-dimensional case, a photonic crystal is nothing more than a periodic dielectric structure. Bragg mirrors consisting of alternating low- and high-index layers constitute, perhaps, the archetypical example [4

4. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

, 5

5. J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).

]. In particular, quarter-wave stacks (at normal incidence) are the most thoroughly studied in connection with omnidirectional reflection (ODR) [6

6. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–82 (1998). [CrossRef] [PubMed]

11

11. J. Lekner “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A 2, 349–53 (2000). [CrossRef]

].

The introduction of Fibonacci multilayers by Kohmoto and coworkers [12

12. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58, 2436–8 (1987). [CrossRef] [PubMed]

] spurred the interest for both possible optical applications [13

13. C. Schwartz, “Reflection properties of pseudorandom multilayers,” Appl. Opt. 27, 1232–4 (1988). [CrossRef] [PubMed]

] and theoretical aspects of light transmission in aperiodic media [14

14. M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B 42, 3680–9 (1990). [CrossRef]

18

18. E. Maciá, “Exploiting quasiperiodic order in the design of optical devices,” Phys. Rev. B 63, 205421 (2001). [CrossRef]

]. In fact, the possibility of obtaining ODR in quasiperiodic Fibonacci multilayers has been put forward recently [19

19. E. Maciá, “Optical engineering with Fibonacci dielectric multilayers,” Appl. Phys. Lett. 73, 3330–2 (1998). [CrossRef]

23

23. J. W. Dong, P. Han, and H. Z. Wang, “Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals.” Chin. Phys. Lett. 20, 1963–5 (2003). [CrossRef]

].

Underlying all these efforts a crucial question remains concerning whether quasiperiodic Fibonacci multilayers would achieve better performance than usual periodic ones. To answer such a fundamental question one first needs to quantify the idea of ODR performance in a unique manner that permits unambiguous comparison between different structures. Only quite recently a suitable figure of merit has been introduced: the area under the transmittance curve as a function of the incidence angle [24

24. T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A 6, 127–31 (2004). [CrossRef]

]. In this paper, we resort to this concept of area to rank in a consistent way the ODR characteristics of these systems.

2. Quasiperiodic Fibonacci multilayers

A Fibonacci system is based on the recursive relation S 0={H}, S 1={L} and Sj =Sj -1 Sj -2 for j≥2. Here H and L are defined as being two dielectric layers with refractive indices (nH,nL) and thicknesses (dH,dL), respectively. The material H has a high refractive index while L is of low refractive index. The number of layers is given by Fj , where Fj is a Fibonacci number obtained from the recursive law Fj =Fj -1+F j-2, with F 0=F 1=1. For j≥3, the systems S j are known as quasiperiodic.

In order to properly compare the optical response of these systems we will rely on the transfer-matrix technique. The transfer matrixM j for the Fibonacci system S j can be computed as [12

12. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58, 2436–8 (1987). [CrossRef] [PubMed]

]

M0=MH,M1=ML,
Mj=Mj1Mj2,j2.
(1)

The transfer matrix for the single layer H is

MH=(cosβHqHsinβH1qHsinβHcosβH),
(2)

and a analogous expression for L. Here βH =(2π/λ)nHdH cosθH is the layer phase thickness, θH being the angle of refraction, which is determined by Snell law. The wavelength in vacuum of the incident radiation is λ. The parameter qH can be written for each basic polarization (p or s) as

qH(p)=nHcosθncosθH,qH(s)=ncosθnHcosθH,
(3)

where we have assumed that the layer is imbedded in a medium of refractive index n. Henceforth θ will denote the angle of incidence and, for simplicity, the surrounding medium we will supposed to be air (n=1).

Let us consider a N-period finite structure whose basic cell is precisely the Fibonacci multilayer Sj . We denote this system as [Sj ] N and its overall transfer matrix is

Mj(N)=(Mj)N.
(4)

The transmittance Tj(N) is given in terms of Mj(N) as [12

12. M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58, 2436–8 (1987). [CrossRef] [PubMed]

]

𝓣j(N)=4Mj(N)2+2,
(5)

where ‖Mj(N)2 denotes the sum of the squares of the matrix elements.

In the theory of periodic systems it is well established that band gaps appear whenever the trace of the basic period Sj satisfies [4

4. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

]

Tr(Mj)2.
(6)

This should be worked out for both basic polarizations. The trace map is a powerful tool to investigate this condition, especially when the index j is high [25

25. M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: Mapping and escape,” Phys. Rev. Lett. 50, 1870–2 (1983). [CrossRef]

]. In our context, it reads as

Tr(Mj+1)=Tr(Mj)Tr(Mj1)Tr(Mj2).
(7)

This simple recurrence relation allows us to compute easily the band gaps. We quote here the first nontrivial cases, namely, the system S 2={LH} and the first quasiperiodic one S 3={LHL}, respectively:

cosβLcosβHΛLHsinβLsinβH1,
cos(2βL)cosβHΛLHsin(2βL)sinβH1.
(8)

The function Λ LH is

ΛLH=12(qLqH+qHqL),
(9)

which is frequency independent but takes different values for p and s polarizations. However, one can check that, irrespective of the angle of incidence, the following relation for both basic polarizations holds:

ΛLH(p)ΛLH(s)1.
(10)

Due to the restriction (10), whenever Eqs. (8) are fulfilled for p polarization, they are always true also for s polarization. In consequence, the p-polarization bands are more stringent than the corresponding s-polarization ones [23

23. J. W. Dong, P. Han, and H. Z. Wang, “Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals.” Chin. Phys. Lett. 20, 1963–5 (2003). [CrossRef]

].

3. Assessing ODR from quasiperiodic multilayers

We first investigate the range of layer thicknesses for which ODR exists; that is, when condition (6) holds true for all the incidence angles. Although for the simple system S 2 analytic approximations are at hand, the general problem seems to be very involved and we content ourselves with a numerical exploration.

For definiteness, we fix the refractive indices to the values nL=1.75 and nH=3.35 at λ=10 µm. In Fig. 1 we have plotted the regions within which ODR exists for the basic periods S j (with j=2,3,4, 5) in terms of the adimensional thicknesses nLdL /λ and nHdH /λ. Note that the use of these adimensional variables not only simplifies the presentation of the results, but, as dispersion can be neglected, the results apply to more general situations.

Fig. 1. Regions where ODR (for p polarization) occurs for the Fibonacci systems S 2={LH}, S 3={LHL}, S 4={LHLLH}, and S 5={LHLLHLHL}. We have taken nL =1.75 and nH =3.35 at λ=10 µm. The inset identifies the filled ellipses and shows the corresponding bandwidths B calculated according to Eq. (15). The marked points correspond to the minimum area for each one of the systems.

The contours of all these regions are approximately elliptical. For every allowed value of nLdL /λ there are two intervals of values of nHdH /λ where the ODR condition is met. This can be traced back to the explicit form of Eqs. (8) for the band gaps. The ellipses for S 2 are the biggest, which confirms that this simple system has the best range of ODR in terms of nd/λ variables. Note also that the usual Bragg solution with layers of a quarter-wavelength thick at normal incidence, namely

nLdLλ=nHdHλ=14,
(11)

works for S 2, but not for the others.

Fig. 2. Area under the transmittance curve, defined in Eq. (12), as a function of nLdL /λ and nHdH /λ for the system S 2, with the same data as in Fig. 1.

It is worth stressing the fact that the regions of ODR for S2 and S 3 are disjoint. This increases the difficulty of comparison between these systems. On the contrary, all the quasiperiodic multilayers have a significant region of common parameters. In fact, from the system S 6 onwards, all the elliptic contours are essentially the same as for the S 5.

These regions of ODR are not enough to fully quantify the performance of the multilayer. In Ref. [24

24. T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A 6, 127–31 (2004). [CrossRef]

] we have proposed that, once the materials and the wavelength are fixed, the area under the transmittance as a function of the angle of incidence θ

𝓐j(N)=0π2𝓣j(N)(θ)dθ,
(12)

is an appropriate figure of merit for the structure. This is a positive number ranging from 0 (perfect reflection) to π/2 (perfect transmission): the smaller the area, the better the performance as ODR. In Fig. 2 we have plotted this area as a function of nLdL /λ and nHdH /λ for S 2. The area has been computed solely for the points fulfilling the ODR condition, so the abrupt steps give the boundaries of ODR plotted in Fig. 1. However, this function varies significantly in the ODR region.

For the present case, the minimum of the area is reached at

nLdLλ=0.34305,nHdHλ=0.25416.
(13)

While the value of nHdH /λ essentially coincides with the standard solution (11), nLdL /λ differs more than 30 %. The fact that the quarter wavelength solution (11) is not the optimum for ODR was pointed out in Ref. [26

26. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightw. Technol. 17, 2018–24 (1999). [CrossRef]

].

Fig. 3. Regions of ODR for the same Fibonacci multilayers as in Fig. 1 in the plane (nL,nH ) of refractive indices. The curves show the limit of ODR for each stack with the optimum thicknesses marked in Fig. 1.

In Fig. 1 we have marked the points of minimum area for each one of the Fibonacci systems Sj . We see the strong difference for the system S 2 and the quasiperiodic systems Sj with j≥3. In fact, for the latter we can summarize the results saying that the optimum area is reached approximately at the values of the parameters

nLdLλ=18,nHdHλ=14.
(14)

In our view, this is a remarkable result: from the principle of minimum area, we have consistently derived optimum parameters for ODR, which differ a lot from the usual solutions found in the literature.

Of course, the optimum parameters for the system S j do not need to be optimum for [Sj ] N . To elucidate this question, we have computed numerically these optimum values of n LdL /λ and nHdH /λ for different systems and for the same refractive indices as before. In Table 1 we have summarized the corresponding data. For simplicity, we have included only results for [Sj ] N up to 26 layers, since from this number onwards all the thicknesses are fairly stable, while the area tends rapidly to 0, as one would expect from a band gap. We can conclude that the optimum parameters do not depend strongly on the number of layers.

Table 1. Optimum parameters nLdL /λ and nHdH /λ, and the corresponding area, for systems [Sj ] N containing up to 26 layers.

table-icon
View This Table

It is quite clear that all the quasiperiodic systems, irrespective of the index j, behave essentially in the same way as far as ODR is concerned. All the points for these systems fit into a straight line. On the other hand, the periodic system [S 2] N lies on another straight line, but with a better slope. That is, for a given number of layers of the system (and under the hypothesis of optimum thicknesses), the system [LH] N offers better performance than any other.

Of course, one may think that the bandwidth of these systems is different. Sometimes the bandwidth is defined at normal incidence, and then it has been argued that quasiperiodic systems offer fundamental advantages [21

21. D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273–9 (2001). [CrossRef]

]. If we denote by λ long and λ short the longer- and shorter-wavelength edges for given ODR bands (of the basic period), it seems more appropriate to define the ODR bandwidth as [10

10. W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–7 (1999). [CrossRef]

]

B=λlongλshort12(λlong+λshort).
(15)
Fig. 4. Logarithm of the area computed for systems [Sj ] N (with j=2,3,4, 5) as a function of the number of layers.

Note that this is the appropriate definition in our case. Obviously, the parameters chosen for the purpose of comparison must be the ones giving minimum area; i. e., optimum ODR behavior. In fact, we have numerically checked that the parameters giving optimum area offer also a good bandwidth. The values of B have been included in the inset of Fig. 1.

4. Concluding remarks

In summary, we have exploited the idea of minimum area to fully assess in a systematic way the performance of omnidirectional reflectors. Although quasiperiodic systems have attracted a lot of interest due to their unusual physical properties, Bragg reflectors offer the best performance, although not at a quarter-wavelength thick at normal incidence. We believe that the best feature of our approach is that it provides a very clear thread to deal with omnidirectional reflection properties in a systematic way. Our method is general and can be applied to any spectral region.

References and links

1.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–62 (1987). [CrossRef] [PubMed]

2.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–9 (1987). [CrossRef] [PubMed]

3.

A complete and up-to-date bibliography on the subject can be found at http://home.earthlink.net/~jpdowling/pbgbib.html

4.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

5.

J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).

6.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–82 (1998). [CrossRef] [PubMed]

7.

J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?,” Science 282, 1841–2 (1998). [CrossRef]

8.

E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23, 1648–9 (1998). [CrossRef]

9.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–8 (1999). [CrossRef]

10.

W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–7 (1999). [CrossRef]

11.

J. Lekner “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A 2, 349–53 (2000). [CrossRef]

12.

M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. 58, 2436–8 (1987). [CrossRef] [PubMed]

13.

C. Schwartz, “Reflection properties of pseudorandom multilayers,” Appl. Opt. 27, 1232–4 (1988). [CrossRef] [PubMed]

14.

M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B 42, 3680–9 (1990). [CrossRef]

15.

A. Latgé and F. Claro, “Optical propagation in multilayered systems,” Opt. Commun. 94, 389–96 (1992). [CrossRef]

16.

N. H. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B 55, 3543–7 (1997). [CrossRef]

17.

M. S. Vasconcelos and E. L. Albuquerque, “Transmission fingerprints in quasiperiodic dielectric multilayers,” Phys. Rev. B 59, 11128–31 (1999). [CrossRef]

18.

E. Maciá, “Exploiting quasiperiodic order in the design of optical devices,” Phys. Rev. B 63, 205421 (2001). [CrossRef]

19.

E. Maciá, “Optical engineering with Fibonacci dielectric multilayers,” Appl. Phys. Lett. 73, 3330–2 (1998). [CrossRef]

20.

E. Cojocaru, “Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,” Appl. Opt. 406319–26 (2001). [CrossRef]

21.

D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. 198, 273–9 (2001). [CrossRef]

22.

R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, “Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,” Appl. Phys. Lett. 80, 3063–5 (2002). [CrossRef]

23.

J. W. Dong, P. Han, and H. Z. Wang, “Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals.” Chin. Phys. Lett. 20, 1963–5 (2003). [CrossRef]

24.

T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A 6, 127–31 (2004). [CrossRef]

25.

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: Mapping and escape,” Phys. Rev. Lett. 50, 1870–2 (1983). [CrossRef]

26.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,” J. Lightw. Technol. 17, 2018–24 (1999). [CrossRef]

OCIS Codes
(230.4170) Optical devices : Multilayers
(310.6860) Thin films : Thin films, optical properties
(350.2460) Other areas of optics : Filters, interference

ToC Category:
Research Papers

History
Original Manuscript: March 30, 2005
Revised Manuscript: May 11, 2005
Published: May 30, 2005

Citation
A. Barriuso, J. Monzón, Luis Sánchez-Soto, and A. Felipe, "Comparing omnidirectional reflection from periodic and quasiperiodic one-dimensional photonic crystals," Opt. Express 13, 3913-3920 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3913


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References

  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-62 (1987). [CrossRef] [PubMed]
  2. S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-9 (1987). [CrossRef] [PubMed]
  3. A complete and up-to-date bibliography on the subject can be found at <a href="http://home.earthlink.net/~jpdowling/pbgbib.html">http://home.earthlink.net/~jpdowling/pbgbib.html</a>.
  4. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  5. J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).
  6. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, �??A dielectric omnidirectional reflector,�?? Science 282, 1679-82 (1998). [CrossRef] [PubMed]
  7. J. P. Dowling, �??Mirror on the wall: you�??re omnidirectional after all?,�?? Science 282, 1841-2 (1998). [CrossRef]
  8. E. Yablonovitch, �??Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,�?? Opt. Lett. 23, 1648-9 (1998). [CrossRef]
  9. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,�?? Appl. Phys. A 68, 25-8 (1999). [CrossRef]
  10. W. H. Southwell, �??Omnidirectional mirror design with quarter-wave dielectric stacks,�?? Appl. Opt. 38, 5464-7 (1999). [CrossRef]
  11. J. Lekner �??Omnidirectional reflection by multilayer dielectric mirrors,�?? J. Opt. A 2, 349-53 (2000). [CrossRef]
  12. M. Kohmoto, B. Sutherland, and K. Iguchi, �??Localization of optics: Quasiperiodic media,�?? Phys. Rev. Lett. 58, 2436-8 (1987). [CrossRef] [PubMed]
  13. C. Schwartz, �??Reflection properties of pseudorandom multilayers,�?? Appl. Opt. 27, 1232-4 (1988). [CrossRef] [PubMed]
  14. M. Dulea, M. Severin, and R. Riklund, �??Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,�?? Phys. Rev. B 42, 3680-9 (1990). [CrossRef]
  15. A. Latgé and F. Claro, �??Optical propagation in multilayered systems,�?? Opt. Commun. 94, 389-96 (1992). [CrossRef]
  16. N. H. Liu, �??Propagation of light waves in Thue-Morse dielectric multilayers,�?? Phys. Rev. B 55, 3543-7 (1997. [CrossRef]
  17. M. S. Vasconcelos and E. L. Albuquerque, �??Transmission fingerprints in quasiperiodic dielectric multilayers,�?? Phys. Rev. B 59, 11128-31 (1999). [CrossRef]
  18. E. Maciá, �??Exploiting quasiperiodic order in the design of optical devices,�?? Phys. Rev. B 63, 205421 (2001). [CrossRef]
  19. E. Maciá, �??Optical engineering with Fibonacci dielectric multilayers,�?? Appl. Phys. Lett. 73, 3330-2 (1998). [CrossRef]
  20. E. Cojocaru, �??Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,�?? Appl. Opt. 40, 6319-26 (2001). [CrossRef]
  21. D. Lusk, I. Abdulhalim and F. Placido, �??Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,�?? Opt. Commun. 198, 273-9 (2001). [CrossRef]
  22. R. W. Peng, X. Q. Huang, F. Qiu, M. Wang, A. Hu, S. S. Jiang, and M. Mazzer, �??Symmetry-induced perfect transmission of light waves in quasiperiodic dielectric multilayers,�?? Appl. Phys. Lett. 80, 3063-5 (2002). [CrossRef]
  23. J. W. Dong, P. Han, and H. Z. Wang, �??Broad omnidirectional reflection band forming using the combination of Fibonacci quasi-periodic and periodic one-dimensional photonic crystals,�?? Chin. Phys. Lett. 20, 1963-5 (2003). [CrossRef]
  24. T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, �??Optimizing omnidirectional reflection by multilayer mirrors,�?? J. Opt. A 6, 127-31 (2004). [CrossRef]
  25. M. Kohmoto, L. P. Kadanoff, and C. Tang, �??Localization problem in one dimension: Mapping and escape,�?? Phys. Rev. Lett. 50, 1870-2 (1983). [CrossRef]
  26. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, �??All-dielectric one-dimensional periodic structures for total omnidirectional reflection and partial spontaneous emission control,�?? J. Lightw. Technol. 17, 2018-24 (1999). [CrossRef]

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