GOOS-HANCHEN SHIFT PDF

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A Nature Research Journal. Metasurfaces with sub-wavelength features are useful in modulating the phase, amplitude or polarization of electromagnetic fields. While several applications are reported for light manipulation and control, the sharp phase changes would be useful in enhancing the beam shifts at reflection from a metasurface. In designed periodic patterns on metal film, at surface plasmon resonance, we demonstrate Goos-Hanchen shift of the order of 70 times the incident wavelength and the angular shifts of several hundred microradians.

We have designed the patterns using rigorous coupled wave analysis RCWA together with S-matrices and have used a complete vector theory to calculate the shifts as well as demonstrate a versatile experimental setup to directly measure the shifts. The giant shifts demonstrated could prove to be useful in enhancing the sensitivity of experiments ranging from atomic force microscopy to gravitational wave detection. Metasurfaces are 2-dimensional equivalents of metamaterials with features smaller than the wavelength specifically designed to modulate the phase, amplitude or polarization of electromagnetic field.

The reports so far covered designer metasurfaces for flat optics to flexible metasurfaces, nonlinear and active metasurfaces that can be controlled, anisotropic metasurfaces for dispersionless response, miniature cavities, antennas, waveguides, complex mode generation by introducing sub-wavelength features within the diffracting apertures among others covered in recent review articles 1 , 2 , 3 , 4 , 5.

Surfaces can also be designed to have sharp phase changes which effect the reflection of an optical beam. A beam reflected off an interface experiences spatial and angular shifts depending on the polarization and the beam profile.

Angular shifts that depend on the beam profile were recently demonstrated by Merano et al. Artmann calculated the GH shift by combining the Fresnel coefficients with the expressions for phase shift for a plane wave and showed that there is polarization dependence in the lateral shift This has been viewed as a consequence of the presence of an evanescent component of the field beyond the interface. More rigorous formulations have been proposed based on Poynting vector analysis, rigorous integrals, coherence matrices and angular spectrum representations 11 , 12 , 13 , Techniques used for reflection at plane interfaces have been extended to patterned surfaces by Guther and Kleeman who used an integral equation system method with parameterization of the periodic profile to calculate the shift and shape of the beams diffracted from a sinusoidal grating 15 , The angular shifts have not been studied much experimentally 9 , although a theoretical treatment 17 for the beam reflected near the Brewster angle has existed for several decades.

Bretenaker et al. This was further followed by the measurement of shifts at single reflection off metallic surfaces Yin et al. Large shifts have also been demonstrated for self-confined light beams For the recent theoretical and experimental advances in this field, one may consult the review article by Bliokh and Aiello Typically, the GH shift at a plane metallic interface is expected to be comparable to the wavelength of light.

So, for any practical application, ways to enhance the shifts are needed. In general, enhancement is expected when there is a large modulation of the complex amplitude of the reflected field as is the case close to the critical angle in total internal reflection TIR and during the excitation of various surface modes. A recent renewal of interest in beam shifts resulted in the demonstration of weak measurement of shifts 24 , 25 , 26 , as well as applications in sensing and switching 27 , General theoretical models to calculate the shifts for incident beam of any polarization has been presented 29 , as well as a unified model applied to harmonic generation by which the second harmonic beam generated at a metallic interface is also shown to experience the shift The sharply varying angular deviations of the beams which occur at such a resonance have potential applications in sensing 9.

In view of the recent work it would be interesting to design patterned surfaces that are optimum for such giant beam shifts. In the current report, we extend our general formulation of shifts to grating interfaces and angular shifts at surface plasmon resonance.

We then proceed to demonstrate a robust high resolution experimental setup that can directly measure the plasmon enhanced shift of p-polarized beam with respect to an s-polarized reference beam. The optical properties of these gratings were calculated using rigorous coupled wave analysis RCWA 31 , 32 , These calculations provide insight into the dispersion relation of various SPP modes that exist in the structure. The specular reflectivity from the grating exhibits a strong dip associated with the SP resonance as shown in Fig.

RCWA has been used to calculate the reflected and transmitted field amplitudes in the air-grating interface given the complex field amplitude of incident plane wave and its polarization.

The optical properties of the fabricated grating metasurfaces Fig. The dark band corresponds to the excitation of a SP mode at the interface. The solid line shows the reflectivity calculated using RCWA.

It manifests itself as a displacement of the centroid of the reflected field distribution with respect to the incident field at the interface. The geometry of the problem is schematically shown in Fig. In order to estimate the shift in the beam, we compute the location of this displaced centroid. This can be evaluated as follows. The incident beam is first resolved into its plane wave components.

The total shift, however, involves contributions from angular deviations which result in significant displacements of the beam centroid as the reflected beam propagates. Here, and , where. Such a field distribution is described by the following expression,.

Also, as a consequence of oblique incidence. The reflected field amplitudes can also be expressed in the form of Eq. For a simple planar interface, one could use the Fresnel reflection formulae to obtain these reflected field amplitudes from the incident amplitudes. Transfer matrices provide a route of computation for stratified media and RCWA can be used for periodically patterned interfaces.

A full treatment of the spatial and angular shifts is presented in the supplementary information which results in the following expression that includes both the spatial and angular shifts. The above expression for the shift in the centroid of the beam in the plane of incidence consists of two parts.

Each of them is estimated by numerical integration. This is in agreement with the intuitive understanding that the shifts tend to be largest when there is a steep variation of the phase of the reflected field with respect to the angle of incidence.

This typically occurs at the surface plasmon resonance condition for interfaces involving metals. However, there can be deviations for focused beams at sharp resonances see Fig. The second part in Eq. This occurs if there exists a reflectivity gradient across the k-space spread of the incident beam. As a result, for a beam with an angular spectrum of finite width, plane wave components corresponding to different angles of incidence experience different reflectivities within the beam.

This results in a change in the direction of the reflected beam. This is more pronounced in situations where the change in reflectivity is sharp such as at a surface plasmon resonance.

Such effects have been discussed by Guther and Kleemann 15 , 16 and also demonstrated experimentally by Merano et al. The displacement in the position of the beam centroid due to the angular deviation is proportional to the distance of propagation of the reflected beam. On propagating a sufficient distance, the angular contribution to the beam displacement tends to overwhelm the spatial GH shift contribution Fig.

The most common way to measure GH shift has been to periodically switch between s- and p- polarization and measure the difference in position using a Quadrant Photodiode QPD and lock-in amplifier combination. While employing this method, however, one must be very careful that the reflected intensities for s- and p- polarizations are equal.

If not, the signal will also include a contribution from the intensity difference between the s- and p- polarized reflections.

Therefore, it would no longer be a faithful measurement of the beam shift. To overcome this limitation, Li et al. We propose a novel measurement technique with high sensitivity as shown in the schematic of Fig. In this technique, a position sensitive detector QPD is scanned across the cross section of the beam around its centroid and the difference signal from the detector is recorded at different detector positions. The measurements were carried out for both s- and p- polarizations simultaneously without otherwise altering the experimental configuration.

By fitting a straight line to the data points measured in such a scan, one can accurately determine the x-intercept, which is the position of the beam centroid. Therefore, difference in the x-intercepts of the two straight lines is the difference between the GH shift for s- and p- polarizations see Fig. The advantage of using such an approach is two-fold. Close to the centroid, the signal from the QPD is small and electronic noise can affect the measurement significantly.

Away from the centroid, however, the electronic noise is much smaller than the QPD signal. By extracting the x-intercept point from the straight line fit to the data, the accuracy of the measurement is considerably improved. Secondly, as detector is scanned across the centroid of the beam, changes due to variation of and even difference between s- and p- incident intensity on the detector only affect the slope of the straight line fits and not the position of the intercepts. Therefore, one does not need to calibrate the response of the QPD with respect to the incident intensity.

This method has been found to be robust and insensitive to variations in the intensity. The method of obtaining the x-intercepts using a straight line fit has been illustrated. The shaded yellow region corresponding to the separation between the x-intercepts of s- and p-polarized beams indicates the relative shift between the s- and p- polarizations.

As shown in Fig. A half-wave plate followed by a polarizer have been used to control the intensity and linear polarization of light being used. The horizontal and vertical polarization components are then split using a polarizing beamsplitter PBS.

The two components are chopped at different frequencies using a double slot chopper wheel. The beam waist is located at the air-grating interface. Using different modulation frequencies for the two polarizations, the centroid positions of both the s- and p- polarizations can be detected in a single scan of the detector. This has proved to be very effective in reducing the duration of data acquisition as well as in reducing the effects of beam drift if any during the measurements.

The experimental setup also allows for measuring the beam intensity profiles using a camera, by blocking either the s- or p- polarized beams as required. Experimental measurements of the shift, using this apparatus, are shown in Fig.

These measured shifts show good agreement with theoretical predictions straight lines after incorporating the effect due to the angular deviation. To illustrate the effects of the angular deviations and propagation of the reflected beam, measurements of the shift have been carried out at different distances from the grating.

These features cause significant changes in the reflected beam centroid position when the beam is allowed to propagate over larger distances. It is interesting to note that above a certain propagation distance, there exists a point where the beam position matches with the the position of the reflected beam spot expected in case of shift-free Euclidean reflection.

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Goos-Hänchen effect in microcavities

The shift is perpendicular to the direction of propagation in the plane containing the incident and reflected beams. This effect is the linear polarization analog of the Imbert—Fedorov effect. This effect occurs because the reflections of a finite sized beam will interfere along a line transverse to the average propagation direction. As shown in the figure, the superposition of two plane waves with slightly different angles of incidence but with the same frequency or wavelength is given by.

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Goos–Hänchen effect

Foster, J. Cook, Opt. My main goal here is to give a basic informal introduction to the phenomenon that forms the basis of our paper. At the time of this writing, this page certainly seems to be more explanatory than the Wikipedia entry. Could I edit the Wikipedia page? Yes, but so could you after reading this page

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Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces

Curator: Paul R. Eugene M. Paul R. Berman , University of Michigan.

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