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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 10 — May. 12, 2008
  • pp: 7580–7586
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The transverse magnetic reflectivity minimum of metals

I. R. Hooper, J. R. Sambles, and A. P. Bassom  »View Author Affiliations


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

The reflectivity of metals has long been a subject of scientific research. Recently there has been a resurgence of interest in this area. In the visible domain this has arisen primarily as a result of Ebbesen and coworkers observation of strongly enhanced transmission through holey metal films [1

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 391, 667–669 (1998). [CrossRef]

]. Further interest has also been stimulated by Pendry’s suggestion of using metals as perfect lenses [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

] while at much the same time the same author [3

3. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

] has re-stimulated general interest in the idea of negative index materials and consequentially the possibility of ‘cloaking’ [4

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 314, 977–980 (2006). [CrossRef] [PubMed]

]. Another proposal that structured perfect metals may support surface modes [5

5. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

] has also stimulated further work at longer wavelength [6

6. A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005). [CrossRef] [PubMed]

]. It is in the context of this interest in metals, and in particular structured metals, as very interesting electromagnetic materials that we wish here to revisit the well-known problem of the simple reflectivity of metals for transverse magnetic or p-polarised radiation.

One of the best treatments of the reflectivity of metals to be found in any textbook is given by Stratton[7

7. J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).

], although a more comprehensive coverage of the minimum in reflectivity for p-polarised radiation is given by Humphreys-Owen [8

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]

]. The central issues are: (i) What is the minimum p-polarised reflectivity for a metal? and (ii) at what angle of incidence does it occur. In addition, we shall address the question of why the minimum in reflectance at long wavelengths is exactly(√2-1)2.

The general Fresnel amplitude reflection coefficient for p-polarised light incident on a planar interface between a dielectric, relative permittivity ε 1, and a metal, complex relative permittivity εm(=εr+i), is given by:

rp=εmε1cosθ[εmε1sin2θ]12εmε1cosθ+[εmε1sin2θ]12
(1)

If we define α=εm/ε 1 and T=tan2 θ equation (1) can be recast as:

rp=α[α+(α1)T]12α+[α+(α1)T]12=αAα+A

where A=α+(α1)T . From this the reflectivity, Rp=rpr p* may be readily computed. Results for the visible, infra-red and microwave domain are illustrated in Fig. 1.

Fig. 1. p-polarised reflectivity from a planar surface as a function of incident angle (θ) for an interface between air and silver described by a Drude model with ωp=1.32×1016 rad/s and τ=1.45×1014 s at wavelengths of 600 nm, 60 microns and 6 mm. Also shown are data for a wavelength of 600 nm with permittivity values taken from Palik [9] of εr=13.91, εi=0.93. (Inset: reflectivity as a function of tan(θ) [log scale]).

In each case a clear minimum in the reflectivity is visible, with that minimum progressing to higher angles as the wavelength, and consequently the magnitude of the permittivities, is increased.

Here we are interested in exploring in some detail this minimum. To find the properties of this minimum all that is required is to differentiate 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)+*(α-T)=0.

This can re-expressed after substituting in α=αr+i and A=α+(α1)T , as;

(12αrαr2+αi2)T3+3T2(αr2+αi2)(1+T)=0

Finally substituting in the material parameters leads to the cubic:

(12εrε1εr2+εi2)ε12T3+3ε12T2(εr2+εi2)(1+T)=0
(2)

This may be shown to be identical in form to equation 13 in Humphreys-Owen’s paper, with the substitution ε 1=1.

Of course equation (2) normally admits three solutions for T but generally only one of these will be purely real while the remaining two constitute a complex conjugate pair. It is the real one which is of primary interest here as we are looking for a real angle solution.

a. Visible domain, for a metal for which |εr|≫εi, ε1 and εr<0

Now (12ε1εr)T3+3T2(εrε1)2(1+T)0 , in which case one solution is T=εrε1 , which is negative and thus tan θ is imaginary. This solution has some interest however as, it gives:

sinθ=εrεr+ε1 .

Fig. 2. Reflectivity minimum due to surface plasmon (SP) excitation using the Kretschmann-Raether [11] geometry (inset). Light of 600 nm wavelength is incident upon a 50 nm thick silver film (εr=-13.91, εi=0.9255) through a glass prism (n=1.5) with air bounding. The SP is excited at a particular internal angle (measured from the normal to the interface) giving a reflection minimum.

This is the well known surface plasmon [10

10. H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).

] condition (Brewster angle for a metal with pure real permittivity). Though the angle at which the surface plasmon excitation occurs is imaginary it can be excited at real angles if some momentum enhancing method is utilized such as the well known Kretschmann-Raether [11

11. E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

] geometry. In this case a reflection minimum can also occur (Fig. 2), but the physics behind this minimum is very different to that of the pseudo-Brewster angle discussed in this paper.

There are two other solutions which take the form:

T=εr2(ε1εr2ε12)[(εr+ε1)±(εr22εrε1+9ε12)12]

Even with εr<0 one of these two solutions is also negative, leaving only one real solution - the one we seek. Substituting in values of -20 for εr and 1 for 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 εi≈|εr|≫ε1 and εr<0

For most metals in the infra-red region of the spectrum there is a wavelength at which the magnitude of the real part of the permittivity becomes equal to the magnitude of the imaginary part of the permittivity. For the Drude model used here to approximately describe the frequency dependent permittivity of silver this occurs at a wavelength of 27.3 µm, at which |εr|=εi=18300. In this situation Eq. (2) can be approximated as:

T3+3T2(εr2+εi2ε12)(1+T)0

Whose solution is T=(εr2+εi2)ε12=(αα*)12 . The angle of the reflectivity minimum rapidly approaches 90° and Rp=2(22)122+(22)12=44.7% as is shown in Fig. 3.

Fig. 3. The angle (open squares) and reflectivity (solid squares) at the reflectivity minimum as a function of| εr |=εi. The angle of the minimum is observed to rapidly approach grazing incidence, whilst the reflectivity of the minimum asymptotically approaches 44.7%. Inset: The reflection minimum for the case when | εr|=εi=18300 which occurs at a wavelength of 27.3 µm for the Drude model with the parameters used here.

c. Microwave domain, for a metal for which εi≫|εr|≫ε1 or n≈k≫n1

Finally we consider the microwave region of the spectrum, a domain in which metals are often considered as perfect conductors, and as such as perfect reflectors. In this case equation 2 becomes:

T3+3T2(εi2ε12)(1+T)0 ,

whereupon T=εi/ε 1.

Substituting this back into Eq. 1 gives

rp=iαi[iαi+(iαi1)αi]12iαi+[iαi+(iαi1)αi]12=1(i)121+(i)12 ,

which leads directly to Rp=(√2-1)2=17.16%.

For both cases b and c above the reflectivity minimum tends to a very high angle and the reflectivity toward a limit value. In the case of the microwave domain this limit value is given by(√2-1)2, but what is the physics of this limit value? This is the question we now address.

Consider p-polarised radiation of wavevector k 0 incident onto a surface at an angle θ. The incident medium has index n1 (purely real) and the metal has permittivity εm(=εr+i. Define normalized k vectors (k′=k/k 0) such that kz=kzr+kzi=kzr/k 0+kzi/k 0 with normalized (conserved) in plane wavevector kx=n 1 sinθ.

This gives kzr 2-kzi 2=εr-ε 1 sin2 θ and 2kzrkzi=εi. Combining these gives a quadratic equation:

[kzr2]2+[εr+ε1sin2θ]kzr2εi24=0 ,

the real solutions of which are

kzr2=εrε1sin2θ+[εr2+εi2+ε12sin4θ2ε1εrsin2θ]122

and

kzi2=εr+ε1sin2θ+[εr2+εi2+ε12sin4θ2ε1εrsin2θ]122 .

Inside the metal the field is described by E=Ex(x̂,(A+iB)ẑ))eikxxeikzrzekzix , and consequently ∇·E=ikxEx+(A+iB)(ikzr-kzi)Ex=0. Resulting in Ex=0 or ikx+(A+iB)(ikzr-kzi)=0.

The second of these solutions results in the following equations:

A2+B2=[EzEx]2=kx2kzr2+kzi2=kx2kzr2+kzi2 , and tanϕ=BA=kzikzr=kzikzr .

We now require solutions for the field components when | εi || εr | and |εi|≫ε 1.

With kzr2=εrε1sin2θ+εi2 and kzi2=εr+ε1sin2θ+εi2 we obtain

[EzEx]2=A2+B2=ε1sin2θεi
(3)
tanϕ=kzikzr=εi12(1εr2εi)εi12(1+εr2εi)1.
(4)

Thus the phase difference between Ex and Ez in the metal is 45°, the key result, as it means that the incident field in which Ex and Ez 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.

From Eqs. (3) and (4) inside the metal we have EzEx=ε1εisinθeiπ4 , whilst in the incident dielectric medium: Einx=cosθ, Einz=sinθ, Erefx=-Eocosθ, Erefz=Eo sinθ, 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 cosθ-Eo cosθ=Ex or

1Eo=Excosθ.
(5)

Normal D is also conserved, and since D=ε E and, in the wavelength range of interest, εmi, we can write ε1sinθ+ε1Eosinθ=εmEziεiEz=iεiε1inθExeiπ4 which, upon rearranging, gives

1+Eoi(εiε1)12Exeiπ4.
(6)

Combining Eq. (5) and (6) leads to

Eoi(εiε1)12cosθeiπ41i(εiε1)12cosθeiπ4+1=iCeiπ41iCeiπ4+1.

Then the reflected intensity is

Iref=(iCeiπ41iCeiπ4+1)(iCeiπ41iCeiπ4+1)=C2+12CC2+1+2C.

It is now a simple matter to find the minimum with respect to C, which occurs when C 2=1. Taking this solution gives: Iref=(√2-1)2=17.16% as predicted. (Note this also gives |r pmin|=√2-1, and εiε1cosθ=1 , which can be rewritten as θεiε1 and is the same as was obtained when solving the cubic.) The minimum value of 17.16% and the dependence of the angle of the minimum on εi is demonstrated in Fig. 4 in which the reflectivity curves for various values of εi are plotted as functions of tan θ (log scale).

Fig. 4. Reflectivity curves for various values of εi as functions of tanθ (log scale) with ε 1=1 and εr=-40000 (such that for the curve with εi=1×107 the system approximates that for silver at a wavelength of 1cm). The asymptotic limit of the minimum reflectance of 17.16% when εi≫|εr|≫ε 1 is clearly evident. Inset: tan(θ) (log scale) as a function of εi (log scale) demonstrating the validity of the equation obtained for the angle of the reflectance minimum θεiε1 in this limit.

In summary we have drawn attention to the curious minimum in the p-polarised reflectivity of metals which, at long wavelengths, gives an elegant solution for the reflectivity of (√2-1)2. It has been shown that, because of the 45° phase difference between the normal and tangential components of E 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)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 391, 667–669 (1998). [CrossRef]

2.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

3.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

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 314, 977–980 (2006). [CrossRef] [PubMed]

5.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

6.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005). [CrossRef] [PubMed]

7.

J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).

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]

9.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

10.

H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).

11.

E. Kretschmann and H. Raether, “Radiative decay of non-radiative surface plasmons excited by light,” Z. Naturforsch. Teil A 23, 2135–2136 (1968).

12.

M. C. Hutley and D. Masytre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976). [CrossRef]

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

  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 391, 667-669 (1998). [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  3. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
  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 314, 977-980 (2006). [CrossRef] [PubMed]
  5. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
  6. A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
  7. J. A. Stratton, Electromagnetic Theory (McGraw Hill, 1941).
  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]
  9. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  10. H. Raether, Surface Plasmons on Smooth and Rough surfaces (Springer-Verlag, 1988).
  11. E. Kretschmann and H. Raether, "Radiative decay of non-radiative surface plasmons excited by light," Z. Naturforsch. Teil A 23, 2135-2136 (1968).
  12. 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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