Abstract

An efficient, general optimized method is outlined that achieves antireflective tapers using lossless, non-dispersive dielectrics. The method modifies the derivative of a perfect antireflective wave amplitude distribution rather than the index of refraction distribution. Modifying the derivative of the wave amplitude distribution minimizes the potential index of refraction distributions and ensures perfect antireflection at one frequency, incidence angle, and linear polarization combination. Additional combinations of frequency, incident angle, and linear polarization can be targeted at a particular reflection coefficient within the optimization. After the method is outlined, three examples are shown with one being fabricated and validated at radiofrequencies.

© 2016 Optical Society of America

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References

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  1. H. Bukhari and K. Sarabandi, “Ultra-wideband printed slot antenna with graded index superstrate,” IEEE Trans. Antenn. Propag. 61(10), 5278–5282 (2013).
    [Crossref]
  2. B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
    [Crossref]
  3. S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
    [Crossref]
  4. M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
    [Crossref]
  5. K. Han and C. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials (Basel) 4(1), 87–128 (2014).
    [Crossref]
  6. A. Sure, T. Dillon, J. Murakowski, C. Lin, D. Pustai, and D. Prather, “Fabrication and characterization of three-dimensional silicon tapers,” Opt. Express 11(26), 3555–3561 (2003).
    [Crossref] [PubMed]
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    [Crossref]
  8. D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley & Sons, 2005).
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    [Crossref]
  10. L. Young, “Synthesis of multiple antireflection films over a prescribed band,” J. Opt. Soc. Am. 51(9), 967–974 (1961).
    [Crossref]
  11. R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE44, 31–35 (1956).
    [Crossref]
  12. M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
    [Crossref]
  13. M. Chen, H. C. Chang, A. S. P. Chang, S. Y. Lin, J. Q. Xi, and E. F. Schubert, “Design of optical path for wide-angle gradient-index antireflection coatings,” Appl. Opt. 46(26), 6533–6538 (2007).
    [Crossref] [PubMed]
  14. Y. Zhang, C. Li, and M. Loncar, “Optimal broadband antireflective taper,” Opt. Lett. 38(5), 646–648 (2013).
    [Crossref] [PubMed]
  15. Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
    [Crossref]
  16. K. H. Kim and Q. H. Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).
    [Crossref] [PubMed]
  17. J. D. Jackson, Classical Electromagnetics, 3rd ed. (Wiley & Sons, 1998).
  18. J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” Proceedings of the 1995 IEEE International Conference on Neural Networks, 1942–1948 (1995).
  19. M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
    [Crossref]
  20. B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
    [Crossref]
  21. J. W. Schultz, Measuring Microwave Materials in Free Space (Createspace, 2012).

2015 (1)

B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
[Crossref]

2014 (3)

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

K. Han and C. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials (Basel) 4(1), 87–128 (2014).
[Crossref]

2013 (3)

H. Bukhari and K. Sarabandi, “Ultra-wideband printed slot antenna with graded index superstrate,” IEEE Trans. Antenn. Propag. 61(10), 5278–5282 (2013).
[Crossref]

K. H. Kim and Q. H. Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).
[Crossref] [PubMed]

Y. Zhang, C. Li, and M. Loncar, “Optimal broadband antireflective taper,” Opt. Lett. 38(5), 646–648 (2013).
[Crossref] [PubMed]

2012 (1)

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

2010 (2)

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

2008 (1)

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

2007 (1)

2003 (1)

1996 (1)

1961 (1)

1957 (1)

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Techniques 5(1), 36–43 (1957).
[Crossref]

Bukhari, H.

H. Bukhari and K. Sarabandi, “Ultra-wideband printed slot antenna with graded index superstrate,” IEEE Trans. Antenn. Propag. 61(10), 5278–5282 (2013).
[Crossref]

Burghoorn, M.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Buskens, P.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Chang, A. S. P.

Chang, C.

K. Han and C. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials (Basel) 4(1), 87–128 (2014).
[Crossref]

Chang, H. C.

Chen, F.

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

Chen, M.

Chhajed, S.

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

Dillon, T.

Eberhart, R.

J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” Proceedings of the 1995 IEEE International Conference on Neural Networks, 1942–1948 (1995).

Good, A.

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

Good, B.

B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
[Crossref]

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

Grann, E. B.

Han, K.

K. Han and C. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials (Basel) 4(1), 87–128 (2014).
[Crossref]

Kennedy, J.

J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” Proceedings of the 1995 IEEE International Conference on Neural Networks, 1942–1948 (1995).

Kim, J. K.

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

Kim, K. H.

K. H. Kim and Q. H. Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).
[Crossref] [PubMed]

Klopfenstein, R. W.

R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE44, 31–35 (1956).
[Crossref]

Kniknie, B.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Li, C.

Lin, C.

Lin, S. Y.

Loncar, M.

Mait, J. N.

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

Mirotznik, M.

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

Mirotznik, M. S.

B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

Moharam, M. G.

Murakowski, J.

Park, Q. H.

K. H. Kim and Q. H. Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).
[Crossref] [PubMed]

Prather, D.

Pustai, D.

Ransom, P.

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

Riblet, H. J.

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Techniques 5(1), 36–43 (1957).
[Crossref]

Sarabandi, K.

H. Bukhari and K. Sarabandi, “Ultra-wideband printed slot antenna with graded index superstrate,” IEEE Trans. Antenn. Propag. 61(10), 5278–5282 (2013).
[Crossref]

Schubert, E. F.

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

M. Chen, H. C. Chang, A. S. P. Chang, S. Y. Lin, J. Q. Xi, and E. F. Schubert, “Design of optical path for wide-angle gradient-index antireflection coatings,” Appl. Opt. 46(26), 6533–6538 (2007).
[Crossref] [PubMed]

Schubert, M. F.

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

Shen, Q.

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

Simmons, S.

B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
[Crossref]

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

Sure, A.

van de Belt, R.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

van Deelen, J.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

van Ee, R.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Vroon, Z.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Wikner, D.

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

Xi, J. Q.

Xu, M.

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Young, L.

Zhang, L.

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

Zhang, Y.

Zhao, Y.

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

AIP Adv. (1)

M. Burghoorn, B. Kniknie, J. van Deelen, M. Xu, Z. Vroon, R. van Ee, R. van de Belt, and P. Buskens, “Improving the efficiency of copper indium gallium (Di-)selenide (CIGS) solar cells through integration of a moth-eye textured resist with a refractive index similar to aluminum doped zinc oxide,” AIP Adv. 4(12), 127154 (2014).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008).
[Crossref]

IEEE Trans. Antenn. Propag. (3)

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Broadband antireflective properties of inverse motheye surfaces,” IEEE Trans. Antenn. Propag. 58(9), 2969–2980 (2010).
[Crossref]

B. Good, S. Simmons, and M. S. Mirotznik, “Design of anti-reflection grading using magneto-dielectric materials,” IEEE Trans. Antenn. Propag. 63(11), 4811–4817 (2015).
[Crossref]

H. Bukhari and K. Sarabandi, “Ultra-wideband printed slot antenna with graded index superstrate,” IEEE Trans. Antenn. Propag. 61(10), 5278–5282 (2013).
[Crossref]

IRE Trans. Microwave Theory Techniques (1)

H. J. Riblet, “General synthesis of quarter-wave impedance transformers,” IRE Trans. Microwave Theory Techniques 5(1), 36–43 (1957).
[Crossref]

J. Opt. Soc. Am. (1)

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

Microw. Opt. Technol. Lett. (2)

B. Good, P. Ransom, S. Simmons, A. Good, and M. Mirotznik, “Design of graded index flat lenses with integrated antireflective properties,” Microw. Opt. Technol. Lett. 54(12), 2774–2781 (2012).
[Crossref]

M. S. Mirotznik, B. Good, P. Ransom, D. Wikner, and J. N. Mait, “Iterative design of moth-eye antireflective surfaces at millimeter wave frequencies,” Microw. Opt. Technol. Lett. 52(3), 561–568 (2010).
[Crossref]

Nanomaterials (Basel) (1)

K. Han and C. Chang, “Numerical modeling of sub-wavelength anti-reflective structures for solar module applications,” Nanomaterials (Basel) 4(1), 87–128 (2014).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

PIERS (1)

Y. Zhao, F. Chen, Q. Shen, and L. Zhang, “Optimal design of graded refractive index profile for broadband omnidirectional antireflective coatings using genetic programming,” PIERS 145, 39–48 (2014).
[Crossref]

Sci. Rep. (1)

K. H. Kim and Q. H. Park, “Perfect anti-reflection from first principles,” Sci. Rep. 3, 1062 (2013).
[Crossref] [PubMed]

Other (5)

J. D. Jackson, Classical Electromagnetics, 3rd ed. (Wiley & Sons, 1998).

J. Kennedy and R. Eberhart, “Particle Swarm Optimization,” Proceedings of the 1995 IEEE International Conference on Neural Networks, 1942–1948 (1995).

J. W. Schultz, Measuring Microwave Materials in Free Space (Createspace, 2012).

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley & Sons, 2005).

R. W. Klopfenstein, “A transmission line taper of improved design,” Proc. IRE44, 31–35 (1956).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Perfect antireflection using transverse wave amplitude function.
Fig. 2
Fig. 2 Iterative optimization algorithm.
Fig. 3
Fig. 3 Example 1 low pass filter as a function of wavelength.(top) The index of refraction distributions for example one of the optimized taper and the corresponding Klopfenstein taper. (bottom) The reflection response of the optimized taper and corresponding Klopfenstein taper.
Fig. 4
Fig. 4 Example 2 Oblique Incidence (top) shows the index of refraction profile of the optimized design. (bottom) Transmitted power for all TE incident waves from 5 to 20 µm. Color scale indicates optical power.
Fig. 5
Fig. 5 Illustration of using a periodic array of subwavelength tapered holes to realize a continuous magneto-dielectric grading.
Fig. 6
Fig. 6 Illustration of using a periodic array of subwavelength tapered holes to realize a continuous grading.
Fig. 7
Fig. 7 (a) Illustration experimental radiofrequency focus beam system. (b) Plot of expected reflection power of the optimized taper and the subwavelength textured moth-eye taper with the measured results using the focused beam system of the fabricated part.

Equations (19)

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

E ˜ (x)= a ^ y P(x) e jQ(x) , H ˜ (x)= a ^ Y z (x)P(x) e jQ(x)
n(x,ω)= 4 μ o S 2 P (x) 4 1 ω 2 μ o P(x) 2 P(x) x 2
P(0)= 2 μ o cS n inc ,P(d)= 2 μ o cS n exit , P x (0)= P x (d)=0.
P(x)= 2 μ o cS [ ( 1 n inc 1 n exit )( 2 x 3 d 3 3 x 2 d 2 )+ 1 n inc ]
P(x) x = 2 μ o cS ( 1 n inc 1 n exit )( 6 x 2 d 3 6 x d 2 )
n inc T ={ n inc cos( θ inc )TE n inc / cos( θ inc ) TM n exit T ={ n exit cos( θ exit )TE n exit / cos( θ exit ) TM
cos( θ exit )= 1 ( n inc sin( θ inc ) n exit ) 2
n T (x)= 4 μ o S 2 P T (x) 4 1 ω 2 μ o P T (x) 2 P T (x) x 2
n T (x)={ n(x)cos(θ(x))TE n(x) / cos(θ(x)) TM
cos(θ(x))= 1 ( n inc n(x) sin θ inc ) 2
x ={ xTE ξ=0 x dξ cos (θ(ξ)) 2 TM
d P ˜ ( x ) dx = d P o ( x ) dx m M1 ( A m ( 2 d ) m ( x d 2 ) m +1 ) e A M x
P new (x)= P o ( 0 )+ P o ( d ) P o ( 0 ) P ˜ ( d ) P ˜ ( 0 ) 0 x d P ˜ ( x ) dx dx
n new (x, ω o )= 4 μ o S 2 P new (x) 4 1 ω o 2 μ o P new (x) 2 P new (x) x 2
F=min( max[ 1 2MN j=1 M i=1 N [ | R TE ( f i , θ j ) |+| R TM ( f i , θ j ) | ] ] )
n eff (x)= n b (1+ 2 v f (x)α 1 v f (x)α )
α= n h 2 n b 2 n h 2 + n b 2
v f (x)= πa (x) 2 Λ 2
a(x)= Λ πα ( n eff (x) n b ) 2 1 ( n eff (x) n b ) 2 +1

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