Goos-Hänchen shifts in harmonic generation from metals

V. J. Yallapragada; Achanta Venu Gopal; G. S. Agarwal

doi:10.1364/OE.21.010878

1. Introduction

Goos and Hänchen [1] showed that light beams, undergoing total internal reflection (TIR), experience a shift within the plane of incidence. An early theoretical treatment of this shift is the work of Artmann [2,3]. Later, it has been shown that such a shift occurs during reflection off an absorbing surface as well [4]. The magnitude of the shift is generally quite small, and its magnitude in TIR is usually less than a micrometer for optical wavelengths. Therefore early experiments such as those by Goos and Hänchen, relied on multiple reflections [1]. However, recent technological advances have made it possible to measure the shifts in a single reflection, and there are several such experiments illustrating this phenomenon in case of both patterned and unpatterned surfaces [5–8]. It has been shown that the shifts are larger close to the critical angle of incidence during TIR [5]. Previously predicted negative Goos-Hänchen (GH) shifts [4] have been observed for reflection off metal surfaces for p polarized incident light [6]. Metallic gratings can yield even larger shifts [7] due to excitation of surface plasmons. Also, giant negative shifts have been predicted for reflection from photonic crystals [9]. There are other shifts that can also occur at interfaces, such as the angular shifts [8], and Fedorov-Imbert shifts [10–12], which are of interest in the context of the spin Hall effect of light [13–16]. The light beam is known to carry both orbital and spin angular momentum [16–18] and the spin-orbit coupling is inherent in the vectorial Maxwell equations. When light travels across the interface between two dielectrics then the orbital angular momentum changes which implies change in the polarization so that the total angular momentum is conserved [19]. In addition, polarization is also important because different shifts are polarization dependent.

The Goos-Hänchen (GH) shift is for linearly polarized light and has traditionally been studied in linear optics, based on the phase properties of the Fresnel coefficients at interfaces. Interestingly enough, the GH shifts have not been studied for reflection of harmonics generated at a nonlinear medium, though the reflection and transmission at the boundary of a nonlinear medium was discussed by Bloembergen and Pershan in early sixties, soon after the experimental discovery of second harmonic generation [20]. As mentioned above, metals are quite promising for the study of shifts, therefore we concentrate on shifts of the harmonics generated at a metal surface. Possibility of second harmonic generation (SHG) at the surface of a centrosymmetric metal, due to the existence of a non-zero surface nonlinear susceptibility has been predicted by Jha [21] and Bloembergen et al [22]. This originates from the discontinuity of the normal component of the electric field at the surface, and has been experimentally demonstrated [22]. The separate contributions to the surface nonlinearity from the surface discontinuity (due to changes when moving from bulk to the surface layer) and the field discontinuity (variation of the local field along the surface layer) have been shown theoretically by P. Guyot-Sionnest et al [23]. Consequences of this discontinuity include the possibility of large surface SHG from high refractive index media. These, and many other implications of the surface second order nonlinearities, have been the subject of extensive research. However, to our knowledge, there has been no effort to address the question of the presence of GH like beam shifts in reflection for the generated second harmonic component.

In this article we develop a general theory for the GH shift at an air–metal interface, partly along the lines of the method used by Li [24], which is then extended to the case of shifts (with respect to the fundamental incident beam) of second harmonic light beam generated at a metal surface.

2. GH shift for fundamental beam

The geometry of the problem is shown in Fig. 1. We study the reflection of a beam of finite extent propagating in air, and getting reflected at the smooth surface of a metal (such as Gold). Since the beam is finite in extent, it has associated with it a spread in the wave vector components. We consider a beam of light whose propagation vector (the central value) lies in the x – z plane, without loss of generality. Such a beam can be described by,

E_{I} = \int d k_{x} \int d k_{y} \exp (i k \cdot r) ℰ_{I} (k_{x}, k_{y})

where we have performed a Fourier expansion of the electric field amplitude into its plane wave components. Here,

k = (k_{x}, k_{y}, k_{z})

. We note that the wave vector components are related according to the following,

k_{x}^{2} + ​ k_{y}^{2} + ​ k_{z}^{2} = {(ω / c)}^{2} ε

where ε refers to the complex dielectric function of the medium which can be taken to be 1 for air. At the interface, Eq. (1) can be rewritten as,

Fig. 1 Shown is the geometry of the problem involving reflection of a light beam off the surface of a metal with the quantities of interest. We consider a beam propagating in the x-z plane, without loss of generality, i.e. the central value of k_y = 0.

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{E_{I} |}_{z = 0} = \int d k_{x} \int d k_{y} \exp (i k_{x} x + i ​ k_{y} y) ℰ_{I} (k_{x}, k_{y})

Each Fourier component $ℰ_{I} (k_{x}, k_{y})$ has a unique s and p component associated with it. Such a treatment is beneficial, as it correctly describes beams encountered experimentally. For example, in the geometry we have considered, say a polarizer is placed in the path of the incident beam that produces beam with its electric field vectors in the xz plane throughout the extent of the beam. However, owing to the spread in wave vector space, this does not imply that the entire beam is s or p polarized. The s and p polarization directions depend on the in-plane component of the wave vector $κ = (k_{x}, k_{y}, 0)$ , are given by $u_{s} = z \times κ / κ$ and $u_{p} = (z \times κ) \times k / κ k$ , respectively, z being the unit vector in the positive z direction. In this basis, the electric field can be represented as,

ℰ_{I} (k_{x}, k_{y}) = ℰ_{I s} (k_{x}, k_{y}) u_{s} (k_{x}, k_{y}) + ℰ_{I p} (k_{x}, k_{y}) u_{p} (k_{x}, k_{y})

Using these components it is straightforward to calculate the reflected and transmitted s and p field components using the Fresnel formulae for the air–metal interface. Our next step is to calculate the shift in the reflected beam. The reflected field at (z = 0) is of the form,

E_{R} (ρ) = \int d k_{x} \int d k_{y} \exp (i κ \cdot ρ) ℰ_{R} (k_{x}, k_{y})

where ρ = (x,y,0). The transmitted electric field at (z = 0) can also be expressed in the same way.

E_{T} (ρ) = \int d k_{x} \int d k_{y} \exp (i κ \cdot ρ) ℰ_{T} (k_{x}, k_{y})

The reflected and transmitted fields can also be resolved into s and p components along the lines of Eq. (4). To calculate the GH shift, we calculate the position of the x-coordinate of centroid of the reflected field. This can be expressed as the following integral.

\begin{matrix} 〈 ρ 〉 = \frac{\int d x \int d y [E^{*}_{R} (ρ) \cdot E_{R} (ρ) ρ]}{\int d x \int d y [E^{*}_{R} (ρ) \cdot E_{R} (ρ)]} \\ = \frac{\int d k_{x} \int d k_{y} \sum_{α} [ℰ^{*}_{R α} (k_{x}, k_{y}) \nabla_{κ} ℰ_{R α} (k_{x}, k_{y})]}{\int d x \int d y [ℰ^{*}_{R} (k_{x}, k_{y}) \cdot ℰ_{R} (k_{x}, k_{y})]}; α = s, p \end{matrix}

The s and p components of the reflected electric field are calculated using the Fresnel formulae for the air-metal interface, and in the current notation, can be expressed as $ℰ_{R s} = (k_{z 0} - k_{z m}) / (k_{z 0} + k_{z m}) ℰ_{I s}$ and $ℰ_{R p} = (k_{z 0} ε - k_{z m}) / (k_{z 0} ε + k_{z m}) ℰ_{I p}$ , where k_z0 and k_zm are the z-components of the propagation vector in air and the metal, respectively, calculated using Eq. (2). It may be noted that, k_zm can be purely imaginary for metals (ε is negative and large). As in the case of the incident field, the total reflected field is given by $ℰ_{R} = ℰ_{R s} u_{s} + ℰ_{R p} u_{p}_{r}$ where u_pr is the reflected u_p component. For an incident field whose centroid lies at ρ = (0,0,0), the GH shift is therefore,

〈 x 〉 = i \frac{\int d k_{x} \int d k_{y} [ℰ_{R s}^{*} \frac{d}{d k_{x}} ℰ_{R s} + ​ ℰ_{R p}^{*} \frac{d}{d k_{x}} ℰ_{R p}]}{\int d x \int d y [{| ℰ_{R s} |}^{2} + {| ℰ_{R p} |}^{2}]}

where we have not shown explicit dependence of quantities on k_x and k_y. To make a correspondence with the lateral beam shift calculated by Artmann, we use the quantity D = <x>cos(θ). It can be shown that the value of D calculated from Eq. (8), in the limit of slowly varying phase, reduces to the formula for the GH shift derived by Artmann. Figure 2 compares the GH-shift calculated using Eq. (8) with the Artmann’s formula for the two polarizations.

Fig. 2 The GH Shift D = <x>cos(θ) of the fundamental beam of wavelength 826 nm in reflection. Red corresponds to an input polarization normal to the xz plane, and black corresponds to polarization parallel to the xz plane. For collimated beams of light, these correspond to s and p polarization. Solid lines correspond to those predicted by Artmann’s formula. Dashed line is y = 0 to guide eye.

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3. GH shift for second harmonic generated beam

In metallic media such as gold, the electromagnetic radiation at frequency ω also leads to the nonlinear polarization at the second harmonic frequency 2ω. The second harmonic generated at the surface of a centrosymmetric medium has been derived by Bloembergen, Jha and associates [20,21]. The polarization is of the form,

P_{N L} (r, 2 ω) = γ \nabla [E^{2} (r, ω)] + β E (r, ω) \nabla \cdot E (r, ω)

where in the two parameters

β = e / 8 π m ω^{2}

and

γ = e [1 - ε (2 ω)] / 8 π m ω^{2}

[25] we use the Gold material parameters from Johnson-Christy [26]. We now proceed to show, using a procedure similar to that derived above, that the second harmonic beam is also shifted with respect to the incident fundamental beam in the x – direction similar to the GH shift of the fundamental beam. For the purpose of this paper we will use the form of the second harmonic response at the metal surface derived by Agarwal and Jha [25]. The second harmonic fields in reflection, denoted by

ℱ_{R}

, can be expressed in the form of Eq. (1) as in the case of the incident beam, as follows,

F_{R} (r) = \int d k_{x} \int d k_{y} \exp (i Q_{R} \cdot r) ℱ_{R} (k_{x}, k_{y})

Here, the reflected wave vector of the second harmonic is

Q_{R} = 2 κ + Λ_{R} z

, where

Λ_{R} = - 2 {({(ω / c)}^{2} - κ^{2})}^{(1 / 2)}

. Similarly the component of the second harmonic inside the metal can be described by,

F_{T} (r) = \int d k_{x} \int d k_{y} \exp (i Q \cdot r) ℱ_{T} (k_{x}, k_{y})

Here, we use the wave vector

Q = 2 κ + Λ z

, inside the metal, and

Λ = 2 {({(ω / c)}^{2} ε (2 ω) - κ^{2})}^{(1 / 2)}

. ε(2ω) being the dielectric function of the metal at the second harmonic frequency 2ω. Since we need the shift of the beam in reflection, we shall compute the centroid of the reflected second harmonic field distribution. The reflected second harmonic field amplitudes in k-space are given by,

ℱ_{R s} = \frac{1}{κ (Λ - Λ_{R})} z \cdot [Λ (κ \times A) + ​ (κ \times (z \times B)]

and

ℱ_{R p} = \frac{Q_{R}}{κ (Λ Q_{R}^{2} - Λ_{R} Q^{2})} [Λ κ \cdot (z \times B) + ​ Q^{2} (κ \cdot A)]

The vectors A and B, which determine the second harmonic response, are given by,

A ​ = - \frac{8 π γ i}{ε (2 ω)} (ℰ_{T} \cdot ℰ_{T}) κ

B = \frac{16 π i ω^{2} β}{c^{2}} (ε (ω) - 1) ℰ_{T z} (z \times ℰ_{T})

The transmitted fields at the fundamental frequency are calculated using the Fresnel formulae $ℰ_{T s} = (2 k_{z 0}) / (k_{z 0} + k_{z m}) ℰ_{I s}$ and $ℰ_{T p} = (2 k_{z 0} \sqrt{ε}) / (k_{z 0} ε + k_{z m}) ℰ_{I p}$ . k_z0 and k_zm are in air and the medium, respectively. The reflected field $ℱ_{R}$ has a wave vector $Q_{R} = 2 κ + Λ_{R} z$ associated with it. Along the lines of our derivation of the GH shift, we can show that the second harmonic light is shifted by an amount,

〈 x 〉 = \frac{i}{2} \frac{\int d k_{x} \int d k_{y} [ℱ_{R s}^{*} \frac{d}{d k_{x}} ℱ_{R s} + ​ ℱ_{R p}^{*} \frac{d}{d k_{x}} ℱ_{R p}]}{\int d x \int d y [{| ℱ_{R s} |}^{2} + {| ℱ_{R p} |}^{2}]}

for an incident field centered at the origin. The factor 2 in Eq. (16) arises as the x component of the wave vector associated with second harmonic is 2k_x. Equations (8) and (16) are the main results of our work, which allow us to treat the GH shift of the reflected fundamental beam, and the second harmonic beam in reflection within the same theoretical framework.

The expressions on the right hand side of Eqs. (8) and (16) need to be numerically integrated to compute the shifts. Equation (8) yields results which are in agreement with those predicted by theory and which were recently verified experimentally by Merano et al. [6]. The predictions of Eq. (8) and Eq. (16) are presented in Figs. 2 and 3, respectively, where we have used a wavelength of 826 nm and a spot size of 800 µm, and integrations were performed numerically. In these calculations, the electric field distribution is assumed to take the form of a Gaussian function.

Fig. 3 Predicted GH shift D = <x>cos(θ) for the reflected fundamental (1600nm) and second harmonic light (800 nm) which is generated at the metal surface. Also shown is the shift for fundamental 800nm. The data points are obtained by numerically integrating Eq. (16). Inset shows the zoomed part between 60 and 90 degrees.

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ℰ_{I} (k_{x}, k_{y}) = ℰ_{0} \sqrt{\frac{2}{π σ_{x} σ_{y}}} \exp (\frac{- {(k_{x} - q)}^{2}}{σ_{x}^{2}}) \exp (\frac{- k_{y}^{2}}{σ_{y}^{2}})

The parameters σ_x and σ_y are related to the spot size w₀ of the beam as $σ_{x} / \cos θ = σ_{y} = \sqrt{2} / w_{0}$ [23]. Here q is the central value of the x component of the incident beam wave vector, given by $2 π \sin θ / λ$ . Figure 2 compares the value of the Goos-Hänchen shift obtained from Eq. (8) with that obtained using the Artmann’s formula.

Figure 3 shows the results for the Goos-Hänchen shift in the second harmonic generated at the air-metal interface. The shift is shown for the polarization in the plane of incidence, since the magnitude of the second harmonic generated using an s polarized input is negligible [22]. It should be emphasized that while the magnitude of the SHG depends on the input power, the shifts are independent of the power and the shift is in the same direction as that for the fundamental beam. We have compared the shift for the fundamental beam at 1600nm and the corresponding second harmonic at 800nm with the shift for fundamental at 800nm. By comparing the results from the present model with those predicted by Artmann’s formula, Fig. 2 clearly establishes the validity of the model for shift of fundamental beam. From Eqs. (8) and (16), it may be seen that the difference in the shift of fundamental and second harmonic beams is not just a factor of 2 but the terms E and F also contribute. Especially, the angle dependence will be different.

4. Discussion

It may be noted that the model used for gold is based on free electron picture and accounts for the intraband transition contribution to dielectric constant. However, the interband contribution also needs to be considered for wavelengths shorter than 500nm where such transitions are possible in gold. We have used widely accepted measured material parameters for gold in our calculations [26]. For high dielectric function media such as metals and semiconductors, the surface beta component of the second order susceptibility dominates. The material model used in this work takes into account this surface contribution in calculating the second harmonic generation. The material model may be more generalized (such that it works for all wavelengths including < 500nm) by considering one of the several models available in literature [27–29].

It may be recalled that the GH shifts can be resonantly enhanced by working with geometries where surface plasmons could be excited [30], and at multilayer structures [31], which makes them suitable to be employed as sensors. The second harmonic depends on the square of the transmitted field at the fundamental and hence geometries with surface plasmon resonances are especially attractive. Unlike the single air-metal interface considered here, plasmonic structures are corrugated metal-dielectric structures (patterned metal or patterned dielectric on metal) or planar structures for prism geometries. As metal films in these cases are not free standing, one needs to consider a three layer structure (for example, air-metal-substrate). In the prism geometry the angle of incidence would have to be above the critical angle to couple light to plasmons. Similarly, in structures with patterned metal film on substrate, one needs to launch the field at specific angles to excite the plasmons. In both cases the appropriate Fresnel coefficients need to be used in the present model. It would be interesting to apply the above exact theory to such structures as well as to other nonlinear phenomena at interfaces. This method can be used to calculate the Fedorov - Imbert shifts, and can therefore be a useful tool for studying the spin Hall effect of light in various geometries.

Finally, shifts of fundamental beam are now measurable as demonstrated by several techniques [5–8,13–16]. Second harmonic generated due to the surface nonlinearity of flat metal could be weak for the position sensitive detectors. However, with photomultiplier tube or avalanche photodiode and a high resolution piezo scanner, one may use beam profiling techniques to study the shift of harmonics generated. In plasmonic structures due to large local field enhancement at plasmon resonances, the second harmonic generation could be stronger and may well be detected by position sensitive detectors.

5. Summary

In summary, we have developed a unified theoretical framework in which we can derive the shifts during beam reflection for the fundamental as well as for the second harmonic generated at a metallic surface. We first compared the results obtained by this model with the results obtained by Artmann’s formula which were experimentally verified. We presented the first calculations on shift in the second harmonic beam generated at the metal surface due to surface nonlinearity. Model presented is easy to extend to plasmonic structures as well as to study other shifts like Federov-Imbert shift.

Acknowledgments

GSA thanks the Director, Tata Institute of Fundamental Research for hosting during the summers of 2011 and 2012 which led to this collaboration. AVG and VJY thank Dr S. Dutta Gupta for discussions.

References and links

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Goos-Hänchen shifts in harmonic generation from metals

Abstract

1. Introduction

2. GH shift for fundamental beam

3. GH shift for second harmonic generated beam

4. Discussion

5. Summary

Acknowledgments

References and links

Cited By

Figures (3)

Equations (17)

Optics Express