## Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles

Optics Express, Vol. 17, Issue 14, pp. 11550-11557 (2009)

http://dx.doi.org/10.1364/OE.17.011550

Acrobat PDF (594 KB)

### Abstract

We demonstrate that a series of one-dimensional photonic crystals made of any dielectric materials, with the periods are distributed in a geometrical progression of a common ratio, r<rc (θ,P), where rc is a structural parameter that depends on the angle of incidence, θ, and polarization, P, is capable of blocking light of any spectral range. If an omni-directional reflection is desired for all polarizations and for all incident angles smaller than θo, then r<rc (θo,p), where p is the polarization with the electric field parallel to the plane of incidence. We present simple and formula like expressions for rc, width of the bandgap, and minimum number of photonic crystals to achieve a perfect light reflection.

© 2009 OSA

*n*[1–5

5. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one dimensional photonic crystals by using photonic hetero-structures,” Appl. Phys. Lett. **80(23)**, 4291–4293 (2002).
[CrossRef]

*n*of 1.6 and 0.8 are required, respectively [6

6. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. **67(4)**, 438–447 (1977).
[CrossRef]

7. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11(11)**, 2892–2899 (1994).
[CrossRef]

*n*are even higher [3

3. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science **282(5394)**, 1679–1682 (1998).
[CrossRef]

4. J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. **23(20)**, 1573–1575 (1998).
[CrossRef]

*n*values cannot be realized for applications of light filtering with UV and visible frequencies, as there are no pair of materials that is non-absorptive (in visible or UV) and at the same time exhibiting large Δ

*n*[8]. The demand on the Δ

*n*in the PC is the limiting factor for successful technological applications. Previously, there were several works on how to enlarge the spectral range for omni-directional reflection [5

5. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one dimensional photonic crystals by using photonic hetero-structures,” Appl. Phys. Lett. **80(23)**, 4291–4293 (2002).
[CrossRef]

9. B. Huang, P. Gu, and L. Yang, “Construction of one-dimensional photonic crystals based on the incident angle domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68(4 Pt 2)**, 046601 (2003).
[CrossRef]

5. X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one dimensional photonic crystals by using photonic hetero-structures,” Appl. Phys. Lett. **80(23)**, 4291–4293 (2002).
[CrossRef]

9. B. Huang, P. Gu, and L. Yang, “Construction of one-dimensional photonic crystals based on the incident angle domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68(4 Pt 2)**, 046601 (2003).
[CrossRef]

*n*, with the periods distributed in a geometrical progression of a common ratio,

*r*,

*r*<

*r*(where

_{c}*r*is a structural parameter we introduced), is capable of reflecting light of any spectral range, at any part of the electromagnetic spectrum. The 1D PCs in the series can be fabricated using matured techniques and non-absorptive materials of any refractive indices and, therefore, are of low fabrication cost.

_{c}*P*) and incident angle (

*θ*). Thereafter, omni-directional bandgaps will be considered.

*m*number of 1D PCs with the period for the

*k*-th PC is

*p*(

_{k}*p*

_{k}_{-1}<

*pk*<

*p*

_{k+1}). For a maximum bandgap, let us assume each of these PCs contains two alternating materials of refractive indices,

*n*

_{1}and

*n*

_{2}with quarter wavelength thicknesses,

*p*/(

_{k}n_{2}*n*

_{1}+

*n*

_{2}) and

*p*/(

_{k}n_{1}*n*

_{1}+

*n*

_{2}), respectively. The center wavelength of the bandgap of the

*k*-th PC is

*λ*=

_{k}*p*, where

_{k}/ω_{c}*ω*is the normalized frequency of the bandgap center. The ratio of the width of the bandgap frequencies, Δ

_{c}*ω*, to

*ω*is independent of

_{c}*p*and can be denoted as

_{k}*g*. Both

_{n}*g*and

_{n}*ω*are dependent on

_{c}*P*and

*θ*, and the values can be calculated using the plane wave expansion methodology [10,11

11. G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and M. T. Doan, “One-dimensional anisotropic photonic crystal with a tunable bandgap,” J. Opt. Soc. Am. B **23(1)**, 159–167 (2006).
[CrossRef]

*θ*=0, we have exact analytical equations given by the following expressions [6

6. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. **67(4)**, 438–447 (1977).
[CrossRef]

7. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11(11)**, 2892–2899 (1994).
[CrossRef]

*θ*=0,

*g*and

_{n}*ω*are polarization independent. The lower and upper bandgap edges of the

_{c}*k*-th PC can be denoted as

*λ-,*and

_{k}*λ+,*, respectively. These edges can be written as,

_{k}*g*is small,

_{n}*λ*±,

_{k}≈

*pk*(1±

*g*/2)/

_{n}*ω*.

_{c}*λ*

_{+,k-1}=

*λ*

_{-,k,}from Eq. (3) it can be shown that the periods,

*p*

_{1},

*p*

_{2},

*p*

_{3}, …, must obey a geometrical progression with a common ratio,

*r*, given by,

_{c}*g*

_{n}is small, then for

*θ*=0, it can be shown that

*r*(

_{c}*θ*=0)≈(9

*n*

_{2}+2

*n*

_{1})/(2

*n*

_{2}+9

*n*

_{1}) (with

*n*

_{2}>

*n*

_{1}). Using Eq. (5) and assuming the periods follow a geometrical progression of a common ratio,

*r*, the condition on the wavelength (Eq. (4) can be written as

*r*<

*r*. It is important to remember that the condition,

_{c}*r*<

*r*, is valid only for a particular set of (

_{c}*θ,P*). The question on how the common ratio should be, in order to reflect light in all angles, will be answered in the later part of the paper.

*ω/ω*is slightly bigger than the

_{c}*g*of a PC with an infinite number of unit cells [6

_{n}6. A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. **67(4)**, 438–447 (1977).
[CrossRef]

7. J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A **11(11)**, 2892–2899 (1994).
[CrossRef]

*N*number of unit cells (see Fig. 1),

*r*, obeys

_{N}*r*>

_{N}*r*. Consequently, the condition

_{c}*r*<

*r*, for creation of the large bandgap, is valid even for the PC with the finite number of unit cells.

_{c}*r*<

*r*, the overall bandgap is only determined by the lower edge of the first PC,

_{c}*λ*

_{-,1}, and the upper edge of the last PC,

*λ*

_{+,m}. Using Eq. (3), we can show that the bandgap width to the bandgap center of the hetero-structure,

*g*, to be exactly,

*g*is the ratio of the variation in the period, |

_{p}*p*-

_{m}*p*

_{1}|, to the average period, (

*p*+

_{m}*p*

_{1})/2. Note that if

*g*/4≪1,

_{p}g_{n}*g*≈

*g*+

_{p}*g*. Equation (6) summarizes two important conclusions. The first is that, even if the refractive index modulation is very small (i.e.,

_{n}*g*≈0), a large bandgap is still possible, if the variation in the period (i.e.,

_{n}*g*) is large. The second is well known [2–4

_{p}4. J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. **23(20)**, 1573–1575 (1998).
[CrossRef]

*g*=0), a large bandgap can be created, if the refractive index modulation (i.e.,

_{p}*g*) is large.

_{n}*r*<

*r*for a given pair of materials,

_{c}*θ*, and

*P*, we can create a bandgap of any size and at any part of the electromagnetic spectrum by just controlling the modulation in the period. So, what is the variation in the period and how many PCs are required to achieve the bandgap for an arbitrary spectral region,

*λ*

_{a}to

*λ*

_{b}? Firstly we set

*λ*-,

_{1}=

*λ*

_{a}and

*λ*

_{+,m}=

*λ*

_{b}in Eq. (3), and find the periods of the first and the last PCs of the hetero-structure. Then, assuming the in between periods follow a geometrical progression with the first and last terms are

*p*

_{1}and

*p*, respectively, the common ratio,

_{m}*r*=(

*p*)

_{m}/p_{1}^{1/m-1}, can be found. The condition in Eq. (3) requires

*r*<

*r*and, therefore, the number of required PCs must satisfy,

_{c}*λ*

_{a}to

*λ*. When

_{b}*g*is very small, the hetero-structure must have a large number of PCs and the period variation must be more continuous [i.e.,

_{n}*g*→0,

_{n}*m*→∞ and hence

*r*=(

*p*)

_{m}/p_{1}^{1/m-1}→1].

**67(4)**, 438–447 (1977).
[CrossRef]

*θ*=0, for the PC hetero-structure (blue curve) with

*n*

_{1}=1.45,

*n*

_{2}=1.8,

*p*

_{1}=125 nm,

*θ*=0,

*N*=12, and

*m*=6. For a comparison, the transmission spectrum of a uniform PC (i.e., a constant period), with the bandgap is designed to be the mid of the visible range is plotted as a green curve in Figs. 2(a). All spectrums are obtained using the transfer matrix method [13

13. M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light* (Pergamon Press Ltd., 1999).
[PubMed]

*m*,

*N*, or the refractive index modulation (i.e., Δ

*n*). For example, Figs. 2(b) and 2(c) shows the transmission spectrums for hetero-structures with the specifications as in Fig. 2(a), but

*m*is increased from 6 to 8 and 16, respectively. As we can readily verify from the figures, the transmission of the narrow spikes in the bandgap region when

*m*=8 [Fig. 2(b)] is smaller than those of

*m*=6 [Fig. 2(a)]. When

*m*=16 [Fig. 2(c)], the narrow spikes cannot be seen in the bandgap region.

*θ, θ*<

*θ*

_{o}? To answer this, we first write

*r*as

_{c}*r*(

_{c}*θ,*), and

_{P}*g*as

_{n}*g*(

_{n}*θ,P*). The polarization

*P*can be either

*s*(electric field is perpendicular to the plane of incidence)-polarization or

*p*(electric field is parallel to the plane of incidence)-polarization. In order to achieve a common bandgap for all

*θ*≤

*θ*

_{o}and all polarizations,

*r*should be smaller than the extreme minimum of the function,

*r*(

_{c}*θ,P*). The minimum value can be obtained by finding the first derivative of

*r*(

_{c}*θ,P*) [Eq. (5) with respect to

*g*(

_{n}*θ,P*) as,

*g*(

_{n}*θ,P*)| is always less than 1. Therefore, the minimum of

*r*(

_{c}*θ,P*) occurs when

*g*(

_{n}*θ,P*) is minimum. The minimum of

*g*(

_{n}*θ,P*) is

*g*(

_{n}*θ*) [determined by the band structure of the

_{o},p*p*-polarization] [3

3. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science **282(5394)**, 1679–1682 (1998).
[CrossRef]

4. J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. **23(20)**, 1573–1575 (1998).
[CrossRef]

*p*-polarization becomes zero, then, there is no condition on

*θ*. On the other hand, if the corresponding point is above the light line, then,

_{o}*θ*<Brewster angle,

_{o}*θ*[4

_{B}**23(20)**, 1573–1575 (1998).
[CrossRef]

*θ*

_{o}=

*θ*, we have

_{B}*g*(

_{n}*θ*)=0.

_{o},p*θ*≤

*θ*

_{o}is

*r*<

*r*(

_{c}*θo,p*). With

*r*<

*r*(

_{c}*θo, p*), the band edges for all polarizations and all angles in the range 0<

*θ*<

*θ*will satisfy Eq. (3) and, therefore, the PC hetero-structure will possess a large bandgap at each value of

_{o}*θ*. Once

*r*is fixed, the next thing to do is to appropriately choose the first and the last periods of the hetero-structure based upon the desired spectral range. The first and the last periods of the hetero-structure can be found using Eq. (3) and noting that the overall common bandgap for

*θ*<

*θ*

_{o}is only determined by the band edges of the first and the last PCs when

*θ*=0 and

*θ*=

*θ*, respectively. In most cases,

_{o}*λ*-,

_{1}(

*θ*=0)>

*λ*-,

_{1}(

*θ*=

*θ*

_{o}) and

*λ+*,

_{m}(

*θ*=

_{0})>

*λ*+,

*(*

_{m}*θ*=

*θ*

_{o}) and, hence, the

*p*

_{1}and

*p*can be found using

_{m}*λ*

_{-,1}(

*θ*=0) and

*λ*

_{+,m}(

*θ*=

*θ*

_{o}), respectively [Eq. (3). Once the periods are found, we can follow the same arguments that lead to Eq. (7), to show that

*m*>ln(

*p*)/ln[

_{m}/p_{1}*r*(

_{c}*θo,*)]≈[

_{p}*g*(

_{p}/g_{n}*θ*)]+1.

_{o,p}*r*(

_{c}*θ*

_{o}=90°,

*p*),

*p*, and the smallest integer values of

_{1}, p_{m}*m*for few materials systems in Tables 1 and 2, for omni-directional light blocking, with UV wavelengths on earth surface (200 nm–400 nm) and visible light (380 nm–780 nm), respectively. Optical and UV materials with the refractive indices shown in Tables 1 and 2 are typical materials and readily available [8,12,14

14. J. Asmussen and D. K. Reinhard, *Diamond Films Handbook* (Marcel Dekker Inc., 2002).
[CrossRef]

15. W. C. Tan, S. Kobayashi, T. Aoki, R. E. Johanson, and S. O. Kasap, “Optical properties of amorphous silicon nitride thin-films prepared by VHF-PECVD using silane and nitrogen,” J. Mater. Sci. Mater. Electron. **20(S1)**, 15–18 (2009).
[CrossRef]

*n*

_{1}=1.45 and

*n*

_{2}=2.40 in Table 2, to calculate the transmission spectrums with air as the ambient medium, a substrate with refractive index, 1.45, and

*N*=12. The spectrums are shown in Fig. 3 for

*θ*=0, 30°, 45°, 60°, 75°, and 85°. As we can see from the figure, the omni-directional bandgap exists for the visible range of the electromagnetic spectrum.

*θ*=0, and the upper bandgap edge wavelength of the

*p*-polarization when

*θ*=90° (see Fig. 3). To analyze the effect of the refractive index dispersions [Fig. 4(a)] to the omni-directional bandgap of the hetero-structure with parameters as in Fig. 3, let assume the bandgap region as a region with the transmissions below 0.3. When

*θ*=0, the lower bandgap edge wavelength increases from 377.6 nm (with the dispersion excluded) to 383.9 nm (with the dispersion included). For

*θ*=90°, the upper bandgap edge wavelength of the

*p*-polarization increases from 773.7 nm (with the dispersion excluded) to 776.0 nm (with the dispersion included). Therefore, the width of the omni-directional bandgap reduces from 396.1 nm (with the dispersion excluded) to 392.1 nm (with the dispersion included).

16. A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe, “Integral merit function for broadband omnidirectional mirrors,” Appl. Opt. **46(15)**, 2903–2906 (2007).
[CrossRef]

17. N. Ponnampalam and R. G. DeCorby, “Analysis and fabrication of hybrid metal-dielectric omni-directional Bragg reflectors,” Appl. Opt. **47(1)**, 30–37 (2008).
[CrossRef]

17. N. Ponnampalam and R. G. DeCorby, “Analysis and fabrication of hybrid metal-dielectric omni-directional Bragg reflectors,” Appl. Opt. **47(1)**, 30–37 (2008).
[CrossRef]

*s*and

*p*-polarizations, respectively. The blue curves in these figures represent AAR spectrums with the refractive index dispersions are neglected, while the red curves in these figures represent AAR spectrums with the dispersions in Fig. 4(a) included. As we can see from Figs. 5(a) and 5(b), the spectral range of the omni-directional reflection for the

*s*-polarization is wider than the corresponding range for the

*p*-polarization. This is consistent with the transmission spectrums shown in Fig. 3, where we can find that the bandgap of the

*s*-polarization is always wider than the bandgap of the

*p*-polarization for all

*θ*=0, 30°, 45°, 60°, 75°, and 85°. The AAR spectrums also clearly indicate that the effect of the dispersion is larger for the UV wavelengths compared to the infra-red wavelengths.

*r*, smaller than a maximum value of

*r*. The paper have presented exact expressions for

_{c}*r*, bandgap to mid gap ratio of the PC hetero-structure, and the minimum number of PCs to achieve the desired range of bandgap in single and all angles of incidence. The proposed method can be used to design filters for vast range of applications such as UV filters (i.e., sunglasses, eye safety glasses, UV photography filters) and visible light filters.

_{c}## References and links

1. | M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature |

2. | J. D. Joannopoulus, R. D. Meade, and J. N. Winn, |

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

4. | J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. |

5. | X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one dimensional photonic crystals by using photonic hetero-structures,” Appl. Phys. Lett. |

6. | A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. |

7. | J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A |

8. | R. W. Waynant, |

9. | B. Huang, P. Gu, and L. Yang, “Construction of one-dimensional photonic crystals based on the incident angle domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

10. | K. Sakoda, |

11. | G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and M. T. Doan, “One-dimensional anisotropic photonic crystal with a tunable bandgap,” J. Opt. Soc. Am. B |

12. | V. G. Dmitriev, G. Gurzadayan, and D. N. Nikogosyan, |

13. | M. Born and E. Wolf, |

14. | J. Asmussen and D. K. Reinhard, |

15. | W. C. Tan, S. Kobayashi, T. Aoki, R. E. Johanson, and S. O. Kasap, “Optical properties of amorphous silicon nitride thin-films prepared by VHF-PECVD using silane and nitrogen,” J. Mater. Sci. Mater. Electron. |

16. | A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe, “Integral merit function for broadband omnidirectional mirrors,” Appl. Opt. |

17. | N. Ponnampalam and R. G. DeCorby, “Analysis and fabrication of hybrid metal-dielectric omni-directional Bragg reflectors,” Appl. Opt. |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 7, 2009

Revised Manuscript: May 29, 2009

Manuscript Accepted: June 18, 2009

Published: June 25, 2009

**Citation**

G. Alagappan and P. Wu, "Geometrically distributed one-dimensional photonic crystals for light-reflection in all angles," Opt. Express **17**, 11550-11557 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11550

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### References

- M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000).
- J. D. Joannopoulus, R. D. Meade, and J. N. Winn, Photonic Crystals Molding the Flow of Light (Princeton, 1995).
- Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282(5394), 1679–1682 (1998). [CrossRef]
- J. N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23(20), 1573–1575 (1998). [CrossRef]
- X. Wang, X. Hu, Y. Li, W. Jia, C. Xu, X. Liu, and J. Zi, “Enlargement of omnidirectional total reflection frequency range in one dimensional photonic crystals by using photonic hetero-structures,” Appl. Phys. Lett. 80(23), 4291–4293 (2002). [CrossRef]
- A. Yariv and P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67(4), 438–447 (1977). [CrossRef]
- J. Lekner, “Light in periodically stratified media,” J. Opt. Soc. Am. A 11(11), 2892–2899 (1994). [CrossRef]
- R. W. Waynant, Electro-Optics Handbook (McGraw-Hill, 2000).
- B. Huang, P. Gu, and L. Yang, “Construction of one-dimensional photonic crystals based on the incident angle domain,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(4 Pt 2), 046601 (2003). [CrossRef]
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2005).
- G. Alagappan, X. W. Sun, P. Shum, M. B. Yu, and M. T. Doan, “One-dimensional anisotropic photonic crystal with a tunable bandgap,” J. Opt. Soc. Am. B 23(1), 159–167 (2006). [CrossRef]
- V. G. Dmitriev, G. Gurzadayan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1997).
- M. Born, and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon Press Ltd., 1999). [PubMed]
- J. Asmussen, and D. K. Reinhard, Diamond Films Handbook (Marcel Dekker Inc., 2002). [CrossRef]
- W. C. Tan, S. Kobayashi, T. Aoki, R. E. Johanson, and S. O. Kasap, “Optical properties of amorphous silicon nitride thin-films prepared by VHF-PECVD using silane and nitrogen,” J. Mater. Sci. Mater. Electron. 20(S1), 15–18 (2009). [CrossRef]
- A. G. Barriuso, J. J. Monzón, L. L. Sánchez-Soto, and A. Felipe, “Integral merit function for broadband omnidirectional mirrors,” Appl. Opt. 46(15), 2903–2906 (2007). [CrossRef]
- N. Ponnampalam and R. G. DeCorby, “Analysis and fabrication of hybrid metal-dielectric omnidirectional Bragg reflectors,” Appl. Opt. 47(1), 30–37 (2008). [CrossRef]

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