Goos-Hänchen shift for a rough metallic mirror

We investigate experimentally the dependence of the Goos-Hänchen shift on the surface properties of an air-metal interface. The shift depends on the microscopic roughness of the metal surface but it is insensitive to the large-scale variations associated with surface non-flatness. Both an effective medium model of roughness and the Rayleigh-Rice theory of scattering are used to interpret the observed phenomenon.


Introduction
The Goos-Hänchen (GH) shift [1] is the displacement, with respect to geometrical reflection, of an s or p polarized light beam reflected by a medium with a complex and angledependent reflection coefficient.It has been generally observed in total internal reflection from a dielectric-air interface.Recently we were successful in measuring, for the first time, the GH shift in conventional metallic reflection [2].These measurements were done on an optical-quality flat Au mirror.This raises the question whether the properties of the metal surface can affect the phenomenon.In order to answer this question, we compare here measurements using three different Au mirror substrates: an optical-quality flat substrate (I), a non-flat (but microscopically smooth) substrate (II) and a microscopically rough substrate (III).The measurements are also compared with theory.

Mirror preparation
We will first describe the sample preparation and characterization.All mirrors were coated by LASEROPTIK [3] with an Au film 200 nm thick.This thickness is an order of magnitude larger than the penetration depth (skin depth) of the 820 nm radiation (used in the experiment) in the Au film.The flatness of the Au mirrors is characterized by Twyman-Green interferometry, the roughness is analyzed with two different techniques, atomic force microscopy (AFM) and optical interference microscopy allowing for analysis in different spatial frequency bands [4].Results of the surface characterization of our Au coated mirrors are summarized in Table 1.cm, a λ/20 surface flatness and a 1 nm rms roughness denoted as σ .
Substrate II was obtained by cutting a 10x10 cm piece from ordinary window glass.Its flatness was found to be 3λ.In Fig. 1 we report the interferogram for this measurement.It was obtained with an expanded He-Ne beam laser (λ = 632 nm) over a diameter of 9 cm.Each fringe that appears in the image is the consequence of a λ/2 phase step.The rms roughness of the same mirror was measured to be below 1 nm.These results are not surprising because window glass is produced with the float glass process, i.e. it is not mechanically polished.Substrate III is an optical-quality substrate, the surface of which was artificially roughed with a standard procedure [5]: the deposition of a thin layer of 2 CaF .It is well known that the surface roughness of a 2 CaF thin-film is an increasing function of its thickness.Since Au thin-films reliably replicate the surface features of the substrates on which they are deposited, one can control the roughness of an Au surface by controlling the thickness of the From the AFM images (Fig. 1) of mirror III we could also deduce an autocorrelation function width δ of 39 nm (HWHM).We produced AFM images of different dimensions, from 3x3 μm 2 down to 500x500 nm 2 .In all cases we find the same rms roughness and the same autocorrelation width.Interference microscopy images were obtained with a green LED (λ = 535 nm) but they were less useful because we were limited by optical diffraction.

Set-up
Our experimental set-up is shown in Fig. 2. A collimated gaussian beam at a wavelength of 820 nm is incident at a given angle on the metallic mirror.With a quadrant detector (New Focus, model 2901), denoted as QD, we measure the displacement of the laser beam in the plane of incidence when the polarization of the beam is switched from p to s .This set-up is similar to the one we used in [2] but with an improved spatial resolution.The improvement comes from the replacement of the light source.In our previous experiment we used a laser diode; here we used a temperature controlled near-infrared fiber-pigtailed super-luminescent diode (SLED) that provides 2 mW of cw radiation at a wavelength of 820 nm (InPhenix IPSDD0802).The choice of this light source is dictated by the fact that it has no higher order transverse modes as a diode laser has.For this reason the beam pointing of the light beam was found to be particularly stable.Another advantage of the SLED is its spectral width (20 nm); being a low coherence length source with respect to a diode laser, spurious noise due to possible speckle formation in the optical set-up is reduced.A microscope objective (20X, tube length 160 mm) collimates the beam that leaves the exit facet of the SLED fiber.The 1/e 2 intensity radius R of the collimated beam after the microscope objective is 890 μm.The beam is p polarized by means of a Glan polarizing prism.Subsequently its polarization is switched between p and s at a frequency of 2.5 Hz with a nematic liquid-crystal variable retarder (MeadowLark Optics) driven by a square-wave voltage.After reflection upon the Au mirror, the QD signal is fed into a lock-in amplifier in order to detect the beam displacements.The QD is mounted on linear translation stages (horizontal and vertical movements) that allow for optimal centering on the reflected beam.A nanovoltmeter (Keithley 181) is used to check optimal centering.The Au mirror is mounted on a homemade θ-2θ goniometer, that allows to speed up the acquisition of the experimental curves.

Experimental results
Fig. 3. Goos-Hänchen shift for mirror I (optical-quality substrate).Experimental data are shown as solid dots and the corresponding theoretical curve, for an ideal flat surface, has been derived from [6,2].The open dots show displacements orthogonal to the plane of incidence; the theoretical line in this case indicates zero displacement.
Experimental results for mirror I are presented in Fig. 3.The quantity accessible for experiments is the difference between p D and s D , where p D ( s D ) is the GH shift for a p ( s ) polarized light beam.Having noticed in early experiments a dependence of the GH shift on the beam position on the mirror, after each series of data we took care to rotate the mirror in order to average eventual position-dependent effects.Experimental data are compared with theoretical predictions [6,2] for an ideally flat optical surface.The refractive index for Au at 820 nm is 2 ˆ0.188 5.39 n i = + [7].The agreement between theoretical prediction and experiment is very good.Mirror I is nominally identical to the one measured in [2].The improvement of the spatial resolution of our set-up, by using a SLED, is evident if we compare the errors bar in the two different experiments, where the errors bars represent the standard deviation of the measurements at each angle of incidence.We have also measured the beam displacement in the plane orthogonal to the plane of incidence; in this case there should be no GH shift.Corresponding data (open dots) are reported in the same graph (Fig. 3).We observe on the two channels the same behavior for the error bars.Figure 4 reports our experimental data for mirror II.Again a comparison with the theoretical prediction for an ideally flat surface is made.Once more the agreement between theory and experiment is very good thus showing that the limited flatness has no effect on the GH shift.
In the case of mirror III, as shown in Fig. 5, we find a clear departure of our experimental data (solid dots) from the theoretical predictions for an ideally flat surface (black line in Fig. 5).Our results prove that the microscopic roughness has an influence on the GH effect.The measured shift is enhanced at small angles of incidence and it is smaller than for the ideal surface at grazing incidence.

Theoretical analysis
We now theoretically address the influence of the surface properties of an Au mirror on the GH effect.The GH shift for a s and a p polarized beam is calculated with the usual Artmann formula [8] starting from the expressions of the phases of the reflection coefficients ( r ) for a given surface.It is thus important to understand how the non-ideal surface properties affect the complex-valued reflection coefficients of our mirrors.
We deal here with two mirrors (II and III) with very different surface properties (Fig. 1).In the case of mirror II the height deviations of the surface are greater than the wavelength, as are the local radii of curvature of the surface structure.The radiation scattered by this type of surface can be treated within the Kirchhoff approximation [9].In this case the GH shift  In the case of mirror III, we are concerned with microscopic roughness where the mean height and the correlation length of the irregularities are both much less than the wavelength of light.Under these restrictions an effective medium model can be applied [11,12] or, alternatively, the Rayleigh-Rice theory for optical scattering can be used [9,10].In the first case roughness is modeled as a finite-thickness effective medium replacing the mirror surface [12].The thick rough metallic plate is treated as a plane-parallel two-layer system consisting of a thick slab, describing the inner part of the conductor, covered by a thin uniform layer of thickness h .The thick slab has a dielectric constant 2 2 bulk n ε = .We model the thin outer layer of thickness h with a Maxwell-Garnett effective medium theory i.e. a model representing roughness as voids in an otherwise homogeneous medium (Au in our case).The void fraction decreases the magnitude of the effective dielectric constant through the dependence of the plasma frequency on the electron volume density [12].Especially the change in the real part of the dielectric constant is noticed.The thickness h of the outer layer is chosen to be 2 σ according to our analysis of the AFM images of the rough mirror.From the same images it is also possible to extract the fraction of the effective plane-parallel layer really occupied by the voids that results to be equal to 0.5.The effective medium theory is slightly more successful, than the theory for an ideal flat surface, in reproducing our experimental observations of the GH effect (red line in Fig. 5).
As already said, an alternative route to follow for a micro-rough surface is the Rayleigh-Rice theory.In this case, for the calculation of the reflection coefficients we have used the formulae given in [10].The theory requires the statistical properties of the surface, such as the rms roughness σ and the autocorrelation length δ .They enter in the explicit form of the power spectral density function of the surface, which we assume to be gaussian.Our results are reported in Fig. 5 (blue line).Again, this theory describes the GH effect somewhat better than the theory for an ideal flat surface.

Conclusion
We conclude that the microscopic roughness is relevant for the GH effect while the macroscopic non-flatness is not.Both the effective medium theory and the Rayleigh-Rice theory describe the GH effect at a microscopically rough surface slightly better than the theory for an ideal flat surface but both are far from perfect.It is conceivable that a better agreement can be reached by explicitly taking into account surface plasmons [13].Roughness may indeed provide the momentum matching required to excite surface plasmons.This can be seen from the dispersion relation for surface plasmons for slightly rough surfaces [13,14].The dispersion relation alone, however, is not sufficient to calculate the Goos-Hänchen shift for a rough surface.This requires knowledge of the reflection phase, i.e., we need an expression for the complex reflection coefficient which takes the excitation of surface plasmons into account.Developing a theory which provides such an expression for microscopic roughness is an interesting and challenging project, but outside the scope of this paper.The large-scale variations associated with surface flatness contain only low spatial frequencies that do not couple efficiently to surface plasmons.

Fig. 1 .
Fig. 1.Characterization of Au-coated mirrors.On the left we show the Twyman-Green interferogram measuring the flatness of mirror II.On the right we show an AFM image of mirror III.

Fig. 4 .
Fig. 4. Goos-Hänchen shift for mirror II (window glass substrate).Experimental data are shown as solid dots and the corresponding theoretical curve is for an ideal flat surface.
by large-scale variations which are equivalent to an angular average over the local shape.This is the reason why we do not see a departure from the theoretical expectation for an ideally flat surface in the case of mirror II.#109935 -$15.00USD Received 10 Apr 2009; revised 22 May 2009; accepted 7 Jun 2009; published 15 Jun 2009 (C) 2009 OSA

Fig. 5 .
Fig.5.Goos-Hänchen shift for mirror III (microscopically rough substrate).Experimental data are shown as solid dots.The theoretical black curve is for an ideal flat surface, the red curve is the prediction for the effective medium model, the blue curve refers to Rayleigh-Rice theory of scattering.

Table 1 . Surface properties of the three mirror substrates investigated in this paper, the optical-quality substrate (I), the window glass substrate (II) and the rough substrate (III).
Substrate I is an optical-quality flat made of Duran ceramic glass with a diameter of 10