## Comparing omnidirectional reflection from periodic and quasiperiodic one-dimensional photonic crystals

Optics Express, Vol. 13, Issue 11, pp. 3913-3920 (2005)

http://dx.doi.org/10.1364/OPEX.13.003913

Acrobat PDF (476 KB)

### 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

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

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. 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]

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]

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]

## 2. Quasiperiodic Fibonacci multilayers

*S*

_{0}={

*H*},

*S*

_{1}={

*L*} and

*S*

_{j}=

*S*

_{j}-

_{1}

*S*

_{j}-

_{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

*F*

_{j}, where

*F*

_{j}is a Fibonacci number obtained from the recursive law

*F*

_{j}=

*F*

_{j}-1+

*F*

_{j-2}, with

*F*

_{0}=

*F*

_{1}=1. For

*j*≥3, the systems

*S*

_{j}are known as quasiperiodic.

_{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]

*H*is

*L*. Here

*β*

_{H}=(2

*π/λ*)

*n*

_{H}

*d*

_{H}cos

*θ*

_{H}is the layer phase thickness,

*θ*

_{H}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

*n*. Henceforth

*θ*will denote the angle of incidence and, for simplicity, the surrounding medium we will supposed to be air (

*n*=1).

*N*-period finite structure whose basic cell is precisely the Fibonacci multilayer

*S*

_{j}. We denote this system as [

*S*

_{j}]

^{N}and its overall transfer matrix is

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

^{2}denotes the sum of the squares of the matrix elements.

*S*

_{j}satisfies [4]

*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]

*S*

_{2}={

*LH*} and the first quasiperiodic one

*S*

_{3}={

*LHL*}, respectively:

_{LH}is

*p*and

*s*polarizations. However, one can check that, irrespective of the angle of incidence, the following relation for both basic polarizations holds:

*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

*S*

_{2}analytic approximations are at hand, the general problem seems to be very involved and we content ourselves with a numerical exploration.

*nL*=1.75 and

*nH*=3.35 at

*λ*=10

*µ*m. 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

*n*

_{L}

*d*

_{L}/λ and

*n*

_{H}

*d*

_{H}/

*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.

*n*

_{L}

*d*

_{L}/

*λ*there are two intervals of values of

*n*

_{H}

*d*

_{H}/

*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

*S*

_{2}, but not for the others.

*S*2 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}.

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]

*θ*

*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

*n*

_{L}

*d*

_{L}/

*λ*and

*n*

_{H}

*d*

_{H}/

*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.

*n*

_{H}

*d*

_{H}/

*λ*essentially coincides with the standard solution (11),

*n*

_{L}

*d*

_{L}/

*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]

*S*

_{j}. We see the strong difference for the system

*S*

_{2}and the quasiperiodic systems

*S*

_{j}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

*n L,nH*) plane for which ODR occurs. In Fig. 3 we have plotted the boundary of such a region for the same Fibonacci multilayers as before: above such curves we have the ODR region. It is again the system

*S*

_{2}the first in fulfilling ODR: the onset of the ODR curve is at

*nH*≃2.5, in agreement with previous estimations [11

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

*S*

_{j}do not need to be optimum for [

*S*

_{j}]

^{N}. To elucidate this question, we have computed numerically these optimum values of

*n*

_{L}

*d*

_{L}/

*λ*and

*n*

_{H}

*d*

_{H}/

*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 [

*S*

_{j}]

^{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.

*S*

_{j}]

^{N}tends to zero exponentially with the number of layers. To test such an

*ansatz*, we have plotted the area (in a logarithmic scale) for all these systems. The results are presented in Fig. 4. To avoid clogging the figure with too much symbols, we have restricted ourselves to the systems with basic periods

*S*

_{j}with

*j*=2,3,4, 5, as in previous figures. We think that a simple glance at this figure is enough to decide on the performance of the Fibonacci systems as omnidirectional reflectors.

*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.

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]

*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*have been included in the inset of Fig. 1.

## 4. Concluding remarks

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

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, |

5. | J. Lekner, |

6. | Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science |

7. | J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?,” Science |

8. | E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. |

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 |

10. | W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. |

11. | J. Lekner “Omnidirectional reflection by multilayer dielectric mirrors,” J. Opt. A |

12. | M. Kohmoto, B. Sutherland, and K. Iguchi, “Localization of optics: Quasiperiodic media,” Phys. Rev. Lett. |

13. | C. Schwartz, “Reflection properties of pseudorandom multilayers,” Appl. Opt. |

14. | M. Dulea, M. Severin, and R. Riklund, “Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,” Phys. Rev. B |

15. | A. Latgé and F. Claro, “Optical propagation in multilayered systems,” Opt. Commun. |

16. | N. H. Liu, “Propagation of light waves in Thue-Morse dielectric multilayers,” Phys. Rev. B |

17. | M. S. Vasconcelos and E. L. Albuquerque, “Transmission fingerprints in quasiperiodic dielectric multilayers,” Phys. Rev. B |

18. | E. Maciá, “Exploiting quasiperiodic order in the design of optical devices,” Phys. Rev. B |

19. | E. Maciá, “Optical engineering with Fibonacci dielectric multilayers,” Appl. Phys. Lett. |

20. | E. Cojocaru, “Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,” Appl. Opt. |

21. | D. Lusk, I. Abdulhalim, and F. Placido, “Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,” Opt. Commun. |

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. |

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. |

24. | T. Yonte, J. J. Monzón, A. Felipe, and L. L. Sánchez-Soto, “Optimizing omnidirectional reflection by multilayer mirrors,” J. Opt. A |

25. | M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: Mapping and escape,” Phys. Rev. Lett. |

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. |

**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

- E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-62 (1987). [CrossRef] [PubMed]
- S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486-9 (1987). [CrossRef] [PubMed]
- 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>.
- P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
- J. Lekner, Theory of Reflection (Dordrecht, The Netherlands, 1987).
- 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]
- J. P. Dowling, �??Mirror on the wall: you�??re omnidirectional after all?,�?? Science 282, 1841-2 (1998). [CrossRef]
- E. Yablonovitch, �??Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,�?? Opt. Lett. 23, 1648-9 (1998). [CrossRef]
- 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]
- W. H. Southwell, �??Omnidirectional mirror design with quarter-wave dielectric stacks,�?? Appl. Opt. 38, 5464-7 (1999). [CrossRef]
- J. Lekner �??Omnidirectional reflection by multilayer dielectric mirrors,�?? J. Opt. A 2, 349-53 (2000). [CrossRef]
- M. Kohmoto, B. Sutherland, and K. Iguchi, �??Localization of optics: Quasiperiodic media,�?? Phys. Rev. Lett. 58, 2436-8 (1987). [CrossRef] [PubMed]
- C. Schwartz, �??Reflection properties of pseudorandom multilayers,�?? Appl. Opt. 27, 1232-4 (1988). [CrossRef] [PubMed]
- M. Dulea, M. Severin, and R. Riklund, �??Transmission of light through deterministic aperiodic non-Fibonaccian multilayers,�?? Phys. Rev. B 42, 3680-9 (1990). [CrossRef]
- A. Latgé and F. Claro, �??Optical propagation in multilayered systems,�?? Opt. Commun. 94, 389-96 (1992). [CrossRef]
- N. H. Liu, �??Propagation of light waves in Thue-Morse dielectric multilayers,�?? Phys. Rev. B 55, 3543-7 (1997. [CrossRef]
- M. S. Vasconcelos and E. L. Albuquerque, �??Transmission fingerprints in quasiperiodic dielectric multilayers,�?? Phys. Rev. B 59, 11128-31 (1999). [CrossRef]
- E. Maciá, �??Exploiting quasiperiodic order in the design of optical devices,�?? Phys. Rev. B 63, 205421 (2001). [CrossRef]
- E. Maciá, �??Optical engineering with Fibonacci dielectric multilayers,�?? Appl. Phys. Lett. 73, 3330-2 (1998). [CrossRef]
- E. Cojocaru, �??Forbidden gaps in finite periodic and quasi-periodic Cantor-like dielectric multilayers at normal incidence,�?? Appl. Opt. 40, 6319-26 (2001). [CrossRef]
- D. Lusk, I. Abdulhalim and F. Placido, �??Omnidirectional reflection from Fibonacci quasi-periodic one-dimensional photonic crystal,�?? Opt. Commun. 198, 273-9 (2001). [CrossRef]
- 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]
- 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]
- 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]
- M. Kohmoto, L. P. Kadanoff, and C. Tang, �??Localization problem in one dimension: Mapping and escape,�?? Phys. Rev. Lett. 50, 1870-2 (1983). [CrossRef]
- 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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