## A new approach for broad-band omnidirectional antireflection coatings

Optics Express, Vol. 15, Issue 15, pp. 9614-9624 (2007)

http://dx.doi.org/10.1364/OE.15.009614

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### Abstract

It is shown that broad-band antireflection coatings with extra large angular range can be designed based on the concept of reflectionless potentials. Numerical calculations for inhomogeneous films with or without substrate demonstrate the above capabilities for both TE and TM polarizations. The design possibilities are infinite and the underlying concept does not rely on standard use of quarter wave plates. Suitable inhomogeneous layers on both sides of a lossless thin dielectric film can thus render it invisible.

© 2007 Optical Society of America

## 1. Introduction

*λ*/4 antireflection coatings with refractive index intermediate between those of the medium of incidence and the substrate. The physical principle that enables the operation of a λ/4 plate is the fact that waves reflected back in the medium of incidence from the two interfaces cancel each other in a destructive interference. Clearly for a given polarization and for given angle of incidence, this can be achieved only at a given wavelength. Thus disturbing any parameter like wavelength or the angle of incidence or the polarization can offset the destructive interference leading to finite reflection. Though there have been many schemes with multiple layers or with variable refractive index profiles, most of the antireflection coatings today[3, 4

4. Q. Tanga, S. Ogura, M. Yamasaki, and K. Kikuchi, “Experimental study on intermediate and gradient index dielectric thin films by a novel reactive sputtering method,” J. Vac. Sci. Technol. A **15**, 2670–2672 (1997). [CrossRef]

5. K. Kintaka, J. Nishii, A. Mizutani, H. Kikuta, and H. Nakano, “Antireflection microstructures fabricated upon fluorine-doped *SiO*_{2} films,” Opt. Lett. **26**, 1642–1644 (2001). [CrossRef]

*rigorous and exact*theoretical foundation of reflectionless potentials which were proposed by Kay and Moses [6

6. I. Kay and H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. **27**, 1503–1508 (1956). [CrossRef]

7. H. B. Thacker, C. Quigg, and J. L. Rosner, “Inverse scattering problem for quarkonium systems. I. One-dimensional formalism and methodology,” Phys. Rev. D **18**, 274–286 (1978). [CrossRef]

1. H. Sankur and W. H. Southwell, “Broadband gradient-index antireflection coating for ZnSe,” Appl. Opt. **23**, 2770–2773 (1984). [CrossRef] [PubMed]

2. D. Poitras and J. A. Dobrowolski, “Toward perfect antireflection coatings. 2. Theory,” Appl. Opt. **43**, 1286–1294 (2004). [CrossRef] [PubMed]

## 2. Propagation equations for TE and TM polarized waves and the reflectionless index profiles

*π*=1) stratified medium with the dielectric function varying as

*ε*=

*ε*(

*z*). Initially we consider the case when

*ε*(

*z*)=

*ε*as

_{s}*z*→±∞, though this will be relaxed later to incorporate the effect of a substrate. Any incident plane wave with arbitrary polarization can be considered to be a mixture of two independent polarizations, namely, the TE (transverse electric) or the TM (transverse magnetic). The TE (TM) wave has only one non-vanishing electric (magnetic) field component perpendicular to the plane of incidence (say, xz plane). Assuming a temporal dependance

*e*

^{-iωt}, the propagation Eqs. for the electric field

*x*-component of the vector for wave incident at -∞ at an angle

*θ*and

*ε*(

*z*) profile introducing

*E*and

*V*(

*z*) as

*V*(

*z*) in Eq.(5) is said to be reflectionless[6

6. I. Kay and H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. **27**, 1503–1508 (1956). [CrossRef]

*ε*(

*z*). Since refractive index is given by the square root of the dielectric function, Eq.(3) can be rewritten to yield the corresponding relectionless refractive index profile

*n*(

*z*) as

*ε*(

*z*) in (2), similar feat leading to an Eq. like (6) is not achievable for the TM-waves. Eq.(4) clearly indicates that a change in the angle corresponds to a change in the energy (albeit in a finite domain) in the corresponding quantum problem. It is thus possible to talk about reflectionless dielectric function profiles for all possible angles of incidence for a given wavelength. As will be shown later such omnidirectional ‘total’ transmission exists even for realistic

*i*.

*e*. truncated (finite domain)

*ε*(

*z*) profiles. However, designing a profile that is reflectionless for both TE and TM waves is not possible (compare Eqs.(1) and (2)). Fortunately, as we will see reflectionless profiles for TE waves turns out to be almost reflectionless even for TM -waves for large angular domains. The situation is a bit more involved in case of wavelength dependence. As is clear from Eq.(6) that the index profile

*n*(

*z*) depends on the wavelength. Potential designed to be reflectionless at one wavelength is not reflectionless at other wavelengths. Fortunately again, the deviation from total transmission at lower wavelengths is not significant. We thus found that the dielectric function profiles based on reflectionless potentials can offer flat response almost with total transmission over large angle and wavelength regions.

6. I. Kay and H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. **27**, 1503–1508 (1956). [CrossRef]

**27**, 1503–1508 (1956). [CrossRef]

*N*positive arbitrary constants

*A*1,

*A*2, …

*A*and κ

_{N}_{1},κ

_{2}, …κ

*, are*

_{N}*given*. One then carries out the following steps

^{2}

*and*

_{n}*f*(

_{n}*z*) have the physical meaning of the eigenvalue and eigenfunction of the corresponding Sturm-Liouville problem with

*V*(

*z*) in Eq.(5) representing the reflectionless potential and the matrix

*M*has a non vanishing positive determinant [6

**27**, 1503–1508 (1956). [CrossRef]

*V*(

*z*) is given by

*M*which in turn are determined by the choice of the free parameters A

_{ij}*,κ*

_{i}*. We give some examples-If we consider only one non-vanishing A*

_{i}_{1}and κ

_{1}with

*A*

_{1}=2κ

_{1}then we get

*A*

_{1}=2κ

_{1}, we chose the maximum of (10)

*i*.

*e*.the refractive index at

*z*=0. The potential corresponding to the

*sech*profile in Eq.(10) is usually referred to as Poschl-Teller (PT) potential and has been studied in detail [16

16. N. Kiriushcheva and S. Kuzmin, “Scattering of a Gaussian wave packet by a reflectionless potential,” Am. J. Phys. **66**, 867–872 (1988). [CrossRef]

*A*

_{1},

*A*

_{2}≠0, we get

*A*’s and

*k*’s in (8) and (9). For defining the potential (8), mathematically there are no constraints on them except their reality and non-negativity. However, since the refractive index values are

*very much limited for realistic materials*, one has to exercise great care in choosing the constants, so as not to end up with unphysical values. Besides, any

*engineered inhomogeneous system*needs to be

*finite*in contrast to the profile (6) (or (9)), which is defined on infinite support. It is thus necessary to look at finite profiles and investigate the deviations from truly reflectionless behavior. Finally the thin AR coatings are to be deposited on a substrate. In order to

*implement*the

*substrate effects*we consider the profile built on a smooth hyperbolic-tangent ramp

*n*

_{s1}and

*n*

_{s2}are the refractive indices of the bounding media on the left and right of the inhomogeneous medium, respectively.

## 3. Numerical results and discussions

*n*(

*z*). Once the design wavelength has been chosen, the practical guideline is offered by the profile (10). For example, for a given λ, using the extremal value of

*V*(

*z*), e.g., -2κ

^{2}

_{1}in Eq.(6), one can estimate the value of κ

_{1}using the following Eq.

*n*is the peak value of the refractive index profile corresponding to (10). For example, for λ=1.06

_{max}*µm*,

*n*=1.0,

_{s}*n*=1.65, Eq.(13) yields a rounded value of κ

_{max}_{1}as 5.5

*µm*

^{-1}. Henceforth, assuming the length unit to be micron, we will suppress all the units in the constants. Thus for the simplest reflectionless index profile one has κ

_{1}=5.5 and

*A*

_{1}=2κ

_{1}=11.0. We now discuss the effect of additional three pairs of constants of the four parameter family on this profile. If the eigenvalues κ

*’s are well separated, then the localized profile remains localized, albeit with some distortions. On the other hand, closely spaced eigenvalues lead to profiles with distinct peaks. The values of*

_{j}*A*’s do not affect qualitatively the shape of the profile. Keeping the aforesaid in mind, we choose the parameters as

_{j}*A*

_{1}=11,

*A*

_{2}=

*A*

_{3}=

*A*

_{4}=3.0, κ

_{1}=5.5,κ

_{2}=0.1,κ

_{3}=1.0,κ

_{4}=9.0.

*without or with the substrate*are shown in the insets of Fig. 1. The inhomogeneous film is assumed to occupy a region -3

*µm*≤

*z*≤ 3

*µm*beyond which the left medium is assumed to be air (

*n*=

_{s}*n*1=1.0), while the substrate is assumed to have a refractive index 1.4 (

_{s}*n*2=1.4). We also used the same set of constants but at a different wavelength (

_{s}*λ*=1.55

*µm*) leading to an analogous profile with a larger peak value of refractive index (see inset of Fig. 2). For numerical simulations we use a

*transfer matrix technique*invoking a fine subdivision and a step-wise constant approximation of the smooth profile. We calculate both the angle and the wavelength dependence of the reflection coefficient. The angle (of incidence) dependence of the intensity reflection coefficient

*R*at λ=1.06

*µm*(λ=1.55

*µm*) for the film in absence or presence of the substrate is shown in Figs. ??(a) (2(a)) and ??(b) (??(b)), respectively. The solid (dashed) curves in these Figs. are for the TE (TM) polarization. One can easily note the flat response over a very large angular range for both the polarizations. The substrate, while retaining this feature, evens out the differences in response for the two polarizations. Figs. 1 and 2 also demonstrate the fact that the

*design principle works well for different wavelengths*.

*µm*(λ=1.55

*µm*) are shown in Figs. 3(a) (4(a)) and 3(b) (4(b)). It is important to note that such films exhibit extremely low reflectivity over a very large range of wavelengths, though they are designed at particular wavelengths. We think that such flat response over such large spectral ranges is not achievable with conventional AR coatings based on quarter wave plates. In the same Figs. we show the

*effect of truncation*. The dashed lines in Figs. 3 and 4 are for -2

*µm*≤

*z*≤2

*µm*. It is clear from the comparison that

*truncation has insignificant effect*if the essential features of the inhomogeneity are retained. We carried out calculations with other higher order families of potentials in order to reveal the parameter dependence of the profiles (not shown). Larger number of parameters offer greater flexibility over the profile leading to lower reflection.

*A*

_{1}=11, κ

_{1}=5.5, (ii)

*A*

_{1}=11, κ

_{1}=5.5,

*A*

_{2}=5.5, κ

_{2}=2.75, and (iii)

*A*

_{1}=11, κ

_{1}=5.5,

*A*

_{2}=5.5, κ

_{2}=2.75,

*A*

_{3}=2, κ

_{3}=1 are shown by the solid, dashed and the dotted lines, respectively. It is clear from the Fig 6. that the two parameter example gives much better result than the Poschl-Teller profile in both angle and frequency scans. The three parameter profile offers better performance in the angle scan, while its frequency response slightly lags behind that of the two parameter family. However, upto the design wavelength (in this example 1.06

*µm*), the performances of all the three profiles are almost the same.

*µm*,

*A*

_{1}=11.0,

*A*

_{2}=8.0,

*A*

_{3}=5.5,

*κ*

_{1}=5.5,

*κ*

_{2}=4.0,

*κ*

_{3}=2.25. For comparison we have shown the results without the ramp in the upper panels of the corresponding Figs. While the angle scan for TE polarization for the profile without the ramp is significantly better than that for the TM (see Fig.7(a)), they are almost identical for the film grown on the substrate (Fig. 7(b)).

*µm*≤

*z*≤3

*µm*for s-polarization is shown in Fig.9 (the inset in Fig.9 shows the profile). As expected, an increasing step-size leads to a degradation of the antireflection behavior. A step-size of 0.005

*µm*for such profiles is accurate enough and has been used in all other calculations.

## 4. Conclusions

17. S. Zaitsu, T. Jitsuno, M. Nakatsuka, and T. Yamanaka, “Optical thin films consisting of nanoscale laminated layers,” Appl. Phys. Lett. **80**, 2442–2444 (2002). [CrossRef]

18. A. Kasikov, J. Aarik, H. Mandar, M. Moppel, M. Pars, and T. Uustare, “Refractive index gradients in *TiO*_{2} thin films grown by atomic layer deposition,” J. Phys. D: Appl. Phys. **39**, 54–60 (2006). [CrossRef]

## Acknowledgment

## References and links

1. | H. Sankur and W. H. Southwell, “Broadband gradient-index antireflection coating for ZnSe,” Appl. Opt. |

2. | D. Poitras and J. A. Dobrowolski, “Toward perfect antireflection coatings. 2. Theory,” Appl. Opt. |

3. | Philippe Lalanne and G Michael Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology8, 53–56 (1997). |

4. | Q. Tanga, S. Ogura, M. Yamasaki, and K. Kikuchi, “Experimental study on intermediate and gradient index dielectric thin films by a novel reactive sputtering method,” J. Vac. Sci. Technol. A |

5. | K. Kintaka, J. Nishii, A. Mizutani, H. Kikuta, and H. Nakano, “Antireflection microstructures fabricated upon fluorine-doped |

6. | I. Kay and H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,” J. Appl. Phys. |

7. | H. B. Thacker, C. Quigg, and J. L. Rosner, “Inverse scattering problem for quarkonium systems. I. One-dimensional formalism and methodology,” Phys. Rev. D |

8. | J. F. Schonfeld, W. Kwong, J. L. Rosner, C. Quigg, and H. B. Thacker, “On the convergence of reflectionless approximation to confining potentials,” Ann. Phys. |

9. | P. G. Drazin, et al, |

10. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

11. | U. Leonhardt, “Optical conformal mapping,” Science |

12. | F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling mechanisms of light transmission through metallic films”, Phys. Rev. Lett. |

13. | J. W. Lee, M. A. Seo, J. Y. Sohn, Y. H. Ahn, and D. S. Kim, “Invisible plasmonic meta-materials through impedance matching to vacuum,” Opt. Express |

14. | G. W. Milton and N. A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance”, Proceedings London Royal Society A |

15. | F. Zolla, S. Guenneau, A. Nicolet, and J. B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect,” Opt. Lett. |

16. | N. Kiriushcheva and S. Kuzmin, “Scattering of a Gaussian wave packet by a reflectionless potential,” Am. J. Phys. |

17. | S. Zaitsu, T. Jitsuno, M. Nakatsuka, and T. Yamanaka, “Optical thin films consisting of nanoscale laminated layers,” Appl. Phys. Lett. |

18. | A. Kasikov, J. Aarik, H. Mandar, M. Moppel, M. Pars, and T. Uustare, “Refractive index gradients in |

**OCIS Codes**

(310.1210) Thin films : Antireflection coatings

(310.1620) Thin films : Interference coatings

**ToC Category:**

Thin Films

**History**

Original Manuscript: April 26, 2007

Revised Manuscript: July 3, 2007

Manuscript Accepted: July 5, 2007

Published: July 18, 2007

**Citation**

S. Dutta Gupta and G. S. Agarwal, "A new approach for broad-band omnidirectional antireflection coatings," Opt. Express **15**, 9614-9624 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9614

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### References

- H. Sankur and W. H. Southwell, "Broadband gradient-index antireflection coating for ZnSe," Appl. Opt. 23, 2770-2773 (1984). [CrossRef] [PubMed]
- D. Poitras and J. A. Dobrowolski, "Toward perfect antireflection coatings. 2. Theory," Appl. Opt. 43, 1286-1294 (2004). [CrossRef] [PubMed]
- P. Lalanne and G M. Morris, "Antireflection behavior of silicon subwavelength periodic structures for visible light," Nanotechnology 8, 53-56 (1997).
- Q. Tanga, S. Ogura, M. Yamasak,i and K. Kikuchi, "Experimental study on intermediate and gradient index dielectric thin films by a novel reactive sputtering method," J. Vac. Sci. Technol. A 15, 2670-2672 (1997). [CrossRef]
- K. Kintaka, J. Nishii, A. Mizutani, H. Kikuta and H. Nakano, "Antireflection microstructures fabricated upon fluorine-doped SiO2 films," Opt. Lett. 26, 1642-1644 (2001). [CrossRef]
- I. Kay and H. E. Moses, "Reflectionless transmission through dielectrics and scattering potentials," J. Appl. Phys. 27, 1503-1508 (1956). [CrossRef]
- H. B. Thacker, C. Quigg, and J. L. Rosner, "Inverse scattering problem for quarkonium systems. I. Onedimensional formalism and methodology," Phys. Rev. D 18, 274-286 (1978). [CrossRef]
- J. F. Schonfeld, W. Kwong, J. L. Rosner, C. Quigg, and H. B. Thacker, "On the convergence of reflectionless approximation to confining potentials," Ann. Phys. 12, 1-28 (1980).
- P. G. Drazin, et al, Solitons- An Introduction, 2nd edition (Cambridge University Press, Cambridge, 1989) Ch.3.
- J. B. Pendry, D. Schurig and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, "Tunneling mechanisms of light transmission through metallic films", Phys. Rev. Lett. 95, 067403 (2005). [CrossRef]
- J. W. Lee, M. A. Seo, J. Y. Sohn, Y. H. Ahn and D. S. Kim, "Invisible plasmonic meta-materials through impedance matching to vacuum," Opt. Express 13, 10681-10687 (2005). [CrossRef] [PubMed]
- G. W. Milton and N. A. Nicorovici, "On the cloaking effects associated with anomalous localised resonance," Proc. R. Soc., London A 462, 3027-3059 (2006). [CrossRef]
- F. Zolla, S. Guenneau, A. Nicolet and J. B. Pendry, "Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect," Opt. Lett. 32, 1069-1071 (2007). [CrossRef] [PubMed]
- N. Kiriushcheva and S. Kuzmin, "Scattering of a Gaussian wave packet by a reflectionless potential," Am. J. Phys. 66, 867-872 (1988). [CrossRef]
- S. Zaitsu, T. Jitsuno, M. Nakatsuka and T. Yamanaka, "Optical thin films consisting of nanoscale laminated layers," Appl. Phys. Lett. 80, 2442-2444 (2002). [CrossRef]
- A. Kasikov, J. Aarik, H. Mandar, M. Moppel, M. Pars and T. Uustare, "Refractive index gradients in TiO2 thin films grown by atomic layer deposition," J. Phys. D: Appl. Phys. 39, 54-60 (2006). [CrossRef]

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