## The transverse magnetic reflectivity minimum of metals

Optics Express, Vol. 16, Issue 10, pp. 7580-7586 (2008)

http://dx.doi.org/10.1364/OE.16.007580

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### Abstract

Metal surfaces, which are generally regarded as excellent reflectors of electromagnetic radiation, may, at high angles of incidence, become strong absorbers for transverse magnetic radiation. This effect, often referred to as the pseudo-Brewster angle, results in a reflectivity minimum, and is most strongly evident in the microwave domain, where metals are often treated as perfect conductors. A detailed analysis of this reflectivity minimum is presented here and it is shown why, in the limit of very long wavelengths, metals close to grazing incidence have a minimum in reflectance given by (√2-1)^{2}.

© 2008 Optical Society of America

## 1. Introduction

8. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. **77**, 949–957 (1961). [CrossRef]

^{2}.

*ε*

_{1}, and a metal, complex relative permittivity

*ε*(=

_{m}*ε*+

_{r}*iε*), is given by:

_{i}*R*=

_{p}*r*

_{p}r_{p}* may be readily computed. Results for the visible, infra-red and microwave domain are illustrated in Fig. 1.

*R*

_{p}with respect to

*θ*or, in this case tan(θ),

*T*, and find the real solutions for which this differential is zero. A little mathematical manipulation leads to the solution:

*A**

*α*(

*α**-

*T*)+

*Aα**(

*α*-

*T*)=0.

*α*=

*α*+

_{r}*iα*and

_{i}*ε*

_{1}=1.

### a. Visible domain, for a metal for which |ε_{r}|≫ε_{i}, ε_{1} and ε_{r}<0

*θ*is imaginary. This solution has some interest however as, it gives:

*ε*<0 one of these two solutions is also negative, leaving only one real solution - the one we seek. Substituting in values of -20 for

_{r}*ε*and 1 for

_{r}*ε*

_{1}gives a

*θ*value of 77°, which agrees with model calculations obtained using Eq. (1).

### b. Infra-red region, for a metal for which ε_{i}≈|ε_{r}|≫ε_{1} and ε_{r}<0

*ε*|=

_{r}*ε*=18300. In this situation Eq. (2) can be approximated as:

_{i}### c. Microwave domain, for a metal for which ε_{i}≫|ε_{r}|≫ε_{1} or n≈k≫n_{1}

*T*=

*ε*/

_{i}*ε*

_{1}.

*R*=(√2-1)

_{p}^{2}=17.16%.

^{2}, but what is the physics of this limit value? This is the question we now address.

*k*

_{0}incident onto a surface at an angle θ. The incident medium has index n

_{1}(purely real) and the metal has permittivity

*ε*(=

_{m}*ε*+

_{r}*iε*. Define normalized

_{i}*k*vectors (

*k*′=

*k*/

*k*

_{0}) such that

*k*′

*=*

_{z}*k*′

*+*

_{zr}*k*′

*=*

_{zi}*k*/

_{zr}*k*

_{0}+

*k*/

_{zi}*k*

_{0}with normalized (conserved) in plane wavevector

*k*′

*=*

_{x}*n*

_{1}sin

*θ*.

*k*′

_{zr}^{2}-

*k*′

_{zi}^{2}=

*ε*-

_{r}*ε*

_{1}sin

^{2}

*θ*and 2

*k*′

*′*

_{zr}k*=*

_{zi}*ε*. Combining these gives a quadratic equation:

_{i}*=*

**E***ik*+(

_{x}E_{x}*A*+

*iB*)(

*ik*-

_{zr}*k*)

_{zi}*E*=0. Resulting in

_{x}*E*=0 or

_{x}*ik*+(

_{x}*A*+

*iB*)(

*ik*-

_{zr}*k*)=0.

_{zi}**|**

*ε*

_{i}**|**≫

**|**

*ε*

_{r}**|**and |

*ε*|≫

_{i}*ε*

_{1}.

*E*and

_{x}*E*in the metal is 45°, the key result, as it means that the incident field in which

_{z}*E*and

_{x}*E*are in phase cannot match the fields inside the metal. Stratton points this out on p523 although it appears to have been largely overlooked. A reflected field is now essential.

_{z}*E*=cos

^{in}_{x}*θ*,

*E*=sin

^{in}_{z}*θ*,

*E*=-

^{ref}_{x}*E*cos

_{o}*θ*,

*E*=

^{ref}_{z}*E*sin

_{o}*θ*, where we assume an incident field of 1 and

*E*

_{0}is the electric field reflected from the interface. The superscripts ‘in’ and ‘ref’ refer to the incident and reflected fields respectively. Since tangential

*E*is conserved we have cos

*θ*-

*E*cos

_{o}*θ*=

*E*or

_{x}*D*is also conserved, and since

**D**=

*ε*

**E**and, in the wavelength range of interest,

*ε*≅

_{m}*iε*, we can write

_{i}*C*

^{2}=1. Taking this solution gives:

*I*=(√2-1)

_{ref}^{2}=17.16% as predicted. (Note this also gives |

*r*

_{pmin}|=√2-1, and

*ε*are plotted as functions of tan

_{i}*θ*(log scale).

^{2}. It has been shown that, because of the 45° phase difference between the normal and tangential components of

*within a metal at these wavelengths, the reflected field can not be zero since the incident field has no out of phase components to match the fields at the boundary. The minimum reflectivity of (√2-1)*

**E**^{2}follows from this 45° phase difference. It should be noted, however, that this minimum limit on the reflectivity is only true for a single interface planar system. If the interface is structured in some manner this minimum value can be lowered.

## References and links

1. | T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature |

2. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

3. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

4. | D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science |

5. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

6. | A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science |

7. | J. A. Stratton, |

8. | S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis,” Proc. Phys. Soc. |

9. | E. D. Palik, |

10. | H. Raether, |

11. | E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A |

12. | M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. |

**OCIS Codes**

(240.0240) Optics at surfaces : Optics at surfaces

(260.3910) Physical optics : Metal optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 31, 2008

Revised Manuscript: May 7, 2008

Manuscript Accepted: May 7, 2008

Published: May 9, 2008

**Citation**

I. R. Hooper, J. R. Sambles, and A. P. Bassom, "The transverse magnetic reflectivity minimum of metals," Opt. Express **16**, 7580-7586 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7580

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### References

- T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
- J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
- J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).
- S. P. F. Humphreys-Owen, "Comparison of reflection methods for measuring optical constants without polarimetric analysis," Proc. Phys. Soc. 77, 949-957 (1961). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
- H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).
- E. Kretschmann and H. Raether, "Radiative decay of non-radiative surface plasmons excited by light," Z. Naturforsch. Teil A 23, 2135-2136 (1968).
- M. C. Hutley and D. Masytre, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431-436 (1976). [CrossRef]

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