The Goos-Hänchen effect for surface plasmon polaritons

Felix Huerkamp; Tamara A. Leskova; Alexei A. Maradudin; Björn Baumeier

doi:10.1364/OE.19.015483

When an electromagnetic beam of finite cross section is incident from an optically more dense medium on its planar interface with an optically less dense medium, and the polar angle of incidence is greater than the critical angle for total internal reflection, the reflected beam undergoes a lateral displacement along the interface, as if it has been reflected from a plane in the optically less dense medium parallel to the physical interface. This effect was first observed by Goos and Hänchen [1], who measured a displacement D = 1.495λ ± 0.261λ for a beam reflected from a silver coated glass-air interface at an angle of incidence θ ₀ = 44.1°. This Goos-Hänchen effect was explained soon after by Artmann [2], who related it to the phase φ(θ ₀) of the amplitude of the reflected beam by

D (θ_{0}) = - \frac{λ}{2 π} \frac{1}{k_{| |} cos θ_{0}} \frac{d φ (θ)}{d θ} |_{θ = θ_{0}} .

In recent years analogues of optical effects originally associated with volume electromagnetic waves have begun to be studied both theoretically and experimentally in the context of surface plasmon polaritons (SPP). These include, e.g. negative refraction [3, 4], the Talbot effect [5, 6], lasing [7], cloaking [8, 9, 10, 11] and Young’s double-slit experiment [12]. The interest in such effects is due to a desire to discover new properties of these surface electromagnetic waves and to the possibility of basing novel nanoscale devices on them.

With these motivations, in this paper we study the analogue of the Goos-Hänchen effect for SPP by investigating the system sketched in Fig. 1 in which a SPP propagating on the surface of a metal whose dielectric function is ɛ ₁(ω) is incident on a planar interface with an optically less dense metal whose dielectric function is ɛ ₂(ω) (|ɛ ₂(ω)| ≫ |ɛ ₁(ω)|). We consider the two cases in which the second metal is either infinitely long (single interface) or of finite length L (double interface). The electromagnetic field of the SPP is determined by use of an impedance boundary condition [13] on the surface x ₃ = 0. Scattering amplitudes A_i(p _||) for p- and s-polarized fields (i = p,s) can be obtained from the solution of a pair of coupled integral equations

A_{i} (p_{| |}) + ζ (ω) \sum_{j = p, s} \int \begin{array}{l} \tilde{S} (p_{| |} - q_{| |}) \frac{M_{i, j} (p_{| |} | q_{| |})}{d_{j} (q_{| |})} A_{j} (q_{| |}) \frac{d^{2} q_{| |}}{{(2 π)}^{2}} \end{array} = - ζ (ω) \tilde{S} (p_{| |} - k_{| |}) M_{i, p} (p_{| |} | k_{| |}) .

Here,

ζ (ω) = \frac{ω}{c} (κ_{2} (ω) - κ_{1} (ω))

with

κ_{j} (ω) = i / \sqrt{- ɛ_{j} (ω)}

and the terms M_{i, j}(p _|||q _||) are given by

\begin{matrix} M_{p, p} (p_{| |} | q_{| |}) & = & - M_{s, s} (p_{| |} | q_{| |}) & = & i {\hat{p}}_{| |} \cdot {\hat{q}}_{| |} \\ M_{p, s} (p_{| |} | q_{| |}) & = & - M_{s, p} (p_{| |} | q_{| |}) & = & {({\hat{p}}_{| |} \times {\hat{q}}_{| |})}_{3} \end{matrix} .

d_{p} (q_{| |}) = β_{0} (q_{| |}) + i \frac{ω}{c} κ_{1} (ω)

and

d_{s} (q_{| |}) = β_{0}^{- 1} (q_{| |}) - i κ_{1} (ω) \frac{c}{ω}

denote the dispersion relations for p- and s-polarization, where

β_{0} (q_{| |}) = \sqrt{q_{| |}^{2} - \frac{ω^{2}}{c^{2}}}

with Reβ ₀(q _||) > 0 and Imβ ₀(q _||)< 0. S̃(Q _|| = p _|| – q _||) is the Fourier transformed of the surface profile function

\tilde{S} (Q_{| |}) = \int_{R^{2}} S (x_{| |}) e^{- Q_{| |} \cdot x_{| |}} d^{2} x_{| |} = 2 π δ (Q_{2}) f (Q_{1}),

where

f^{I} (Q_{1}) = \frac{1}{i (Q_{1} - i η)}, f^{II} (Q_{1}) = \frac{L sinc (Q_{1})}{2 exp (\frac{i Q_{1} L}{2})}

for the single and double interface, respectively. Due to the translational invariance of the system in the x ₂-direction A_p,s(q _||) have the general form

A_{p, s} (q_{| |}) = 2 π δ (q_{2} - k_{2}) a_{p, s} (q_{1}) .

Substituting Eq. (6) into Eq. (2) leads to a pair of effective one-dimensional integral equations

a_{i} (p_{1}) + ζ (ω) \sum_{j = p, s} \int_{- \infty}^{\infty} M_{i, j} ({\bar{p}}_{| |} | {\bar{q}}_{| |}) f (p_{1} - q_{1}) \frac{a_{j} (q_{1})}{d_{j} ({\bar{q}}_{| |})} \frac{d q_{1}}{2 π} = - ζ (ω) M_{i, p} ({\bar{p}}_{| |} | k_{| |}) f (p_{1} - k_{1}),

in which we define p̄ _|| = (p ₁, k ₂) and q̄ _|| = (q ₁, k ₂). Equations (7) are solved numerically using the Nystrom method [14]. The infinite range of integration is replaced by a finite interval [–q _∞,q _∞]. The resulting integrals over q ₁ were converted to sums using a N-point extended midpoint method. p ₁ was given the values of the abscissas used in the evaluation of the integrals and a square 2N × 2N supermatrix equation with N = 18001 for a_p,s(p ₁) is solved by a standard linear equation solver. The convergence of the solution was monitored by increasing q _∞ and N systematically until the solution did not change upon further increases of these parameters. A lateral displacement of the incident SPP beam is identifiable in the far field region by the intensity distribution of the scattered electromagnetic field of the propagating p-polarized SPP mode with wave number k _||. For an incident plane wave, the scattered field is given by

E^{sc} (x | ω) = \int \frac{{\hat{e}}_{p} (q_{| |}) A_{p} (q_{| |})}{β_{0} (q_{| |}) + i \frac{ω}{c} κ_{1} (ω)} e^{i q_{| |} x_{| |} - β_{0} (q_{| |}) x_{3}} \frac{d^{2} q_{| |}}{{(2 π)}^{2}}

where

{\hat{e}}_{p} (q_{| |}) = \frac{c}{ω} {i {\hat{q}}_{| |} β_{0} (q_{| |}) - {\hat{x}}_{3} q_{| |}}

is the polarization vector for p-polarized SPP. The contribution to this field in the region x ₁ < 0 from the reflected surface plasmon polariton is given by the residue of the integrand at the simple pole it has at q ₁ = –k ₁(ω) = –cos(θ)k _||(ω). With the assumption that ɛ ₁(ω) has an infinitesimal positive imaginary part, this pole lies in the lower half of the complex q ₁ plane. It can be shown that a_p(q ₁) has no pole in this region. On evaluating the residue at this pole we obtain for the electric field of the reflected SPP in the region x ₁ < 0, x ₃ > 0

E^{ref} (x | ω) = r (- k_{1}) \frac{c}{ω} {\hat{e}}_{p} (- k_{1}, k_{2}) e^{- i k_{1} x_{1} + i k_{2} x_{2} - β_{0} (k_{| |}) x_{3}},

where

r (- k_{1}) = \frac{ω}{c} κ_{1} \frac{a_{p} (- k_{1})}{k_{1}} = R (- k_{1}) e^{i φ (- k_{1})}

is the reflection amplitude.

Fig. 1 Scattering geometry of the double interface system. The blue (gray) vectors are the beam reflected from the first interface, the red (black) vectors are the beams reflected from the second interface, and the green (light gray) vectors are the actual reflected beams.

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In Fig. 2 we present a plot of R as a function of the angle of incidence when a SPP in the form of a plane wave whose wavelength is λ = 632.8 nm, propagating on a gold surface with ɛ ₁(ω) = −11.8 at the corresponding frequency, is incident on its planar interface with aluminum (ɛ ₂(ω) = −64.07). Since the mean free paths of SPP on these two surfaces are L ₁ = 7μm and L ₂ = 30μm, we expect the effect of ohmic losses on the our results obtained for real-valued ɛ_i(ω) to be small. The angle of incidence is θ = 78°, and the 1/e half width of the beam is w = 20c/ω. The critical angle for total internal reflection, given by $sin θ_{c} = {[1 - 1 / ɛ_{1} (ω)]}^{\frac{1}{2}} / {[1 - 1 / ɛ_{2} (ω)]}^{\frac{1}{2}}$ , has the value θ_c = 75.4° in this case. Note that preliminary, non-converged results were shown unanalyzed as work in progress in Ref. [15].

Fig. 2 The modulus of the reflection amplitude R and phase shift φ as functions of the angle of incidence θ for an incident SPP plane wave at the single interface.

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It is seen from this figure that R is small (∼10⁻⁴) for all angles smaller than θ_c, and equal to unity for angles greater than θ _c. R is not a monotonically increasing function of θ, but has a pronounced minimum at the angle of incidence θ ≈ 45°. The occurrence of this dip has been explained for a somewhat different SPP scattering problem as the Brewster effect for the incident SPP [16]. The shift of the position of the minimum in Fig. 2 from θ = 45° is due to a small imaginary part added to ɛ ₁(ω). The phase shift φ(θ) at the interface is close to π for angles smaller than θ = 45°, and jumps to nearly 2π at this angle. In the rather narrow interval [θ _c, 90°] the phase decreases continuously from 2π to π. Because of the derivative of φ(θ) in Artmann’s result one can therefore expect a significant lateral displacement of the reflected beam.

To observe the Goos–Hänchen effect we need the intensity distribution of the field of the reflected SPP when the incident SPP has the form of a beam instead of a plane wave. Such a field is represented by a superposition of plane waves weighted by a Gaussian function of θ with 1/e half width 2/[k _|| (ω)w], centered at θ = θ ₀, and normalized to unity, which yields a Gaussian beam of 1/e half width w whose angle of incidence is θ ₀.

In Fig. 3 we present a color-level plot of the intensity distribution of the incident and reflected SPP beams for the system assumed in obtaining the results plotted in Fig. 2. The positions of the maxima of both beams at the interface (x ₁ = 0) are marked with a dashed line, showing a displacement of D = 60.3c/ω = 9.6λ.

Fig. 3 (a) Color-level plot of the intensity of the incident beam (left), and the reflected beam (right). The angle of incidence is θ ₀ = 78°, the beam width is w = 20c/ω and x ₃ = 0.1c/ω. The maxima of the incident and reflected beam are marked with dashed lines, the displacement is D = 60.3c/ω = 9.6λ. (b) Plot of the respective intensities along the x ₂-direction at the interface (x ₁ = 0).

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In Fig. 4, we compare the results for D as a function of the angle of incidence of the beam θ ₀ for different widths of the beam. For a broad beam (w = 200 c/ω, solid line) we note the existence of a pronounced negative displacement for an angle of incidence close to 45° arising from the jump in the phase shift from π to 2π as in Fig. 2. However, since the reflectivity R is very small (about 10⁻⁵) at this angle, it might be difficult to observe this negative displacement experimentally. At θ _c, D increases substantially, and the pronounced peak can be related to the maximum slope of φ(θ) in Fig. 2. The behavior for θ ₀ → 90° is in agreement with 1/cos(θ ₀) dependence in Artmann’s formula. The faint negative D close to θ _c is a result of the use of a small value for η = 0.01 in f ^I(Q ₁).

Fig. 4 Calculated lateral displacement D for different beam widths (200, 30, and 10 c/ω) as a function of the angle of incidence of the beam θ ₀.

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In the case of beams with smaller half widths, Artmann’s formula no longer holds, resulting in noticeable differences in the dependence of D on the angle of incidence: the smaller the width, the fewer the structures in D(θ ₀), e.g. as in Fig. 4 for beams with w = 30 c/ω (dashed line) or w = 10c/ω (dotted line). For instance, the negative displacement at θ ₀ = 45° becomes less pronounced and the feature in the curve smears out. Similar observations can be made for the structures close to the critical angle for total internal reflection. In particular, D remains finite when θ ₀ → 90°.

At a double-interface with L = 20c/ω the phase of a reflected SPP plane wave (see inset of Fig. 5) is identical to the one at the single interface in the interval [θ _c, 90°] since the wave is totally reflected at the first interface. The major difference is the oscillating behavior at smaller angles, which is due to multiple scattering at the two interfaces leading to destructive or constructive interference at different angles as in a Fabry-Pérot interferometer. These oscillations also appear in D(θ ₀) shown in Fig. 5 and decay with larger L.

Fig. 5 Lateral displacement as a function of angle of incidence θ ₀ at a double interface with L = 20c/ω (solid line) compared to the result for a single interface (dashed line). The width of the SPP beam is w = 30c/ω. The inset shows the phase of a reflected SPP plane wave as a function of θ.

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The absolute values for the lateral displacements of SPP beams of up to 25λ are about one order of magnitude larger than the corresponding results for volume waves [1]. This can be rationalized by the larger critical angle and the concomitantly steeper decrease of the phase in the interval [θ _c, 90°] in the case of SPP. The values are also sensitive to changes in the dielectric functions. Figure 6 shows the change in D upon variation of either ɛ ₁(ω) or ɛ ₂(ω) while the other one is fixed. This results were obtaind by the use of Artmann’s formula (1). In particular, the strong dependence of the calculated lateral displacement on the value of ɛ ₁(ω), i.e. the dielectric function of the gold surface, indicates that small modifications of the latter may be resolved. Adsorption of molecules changes the dielectric environment of surfaces. Experimental measurements of the Goos-Hänchen effect for SPP for θ ₀ > θ _c, i.e., at grazing incidence, depending on molecular coverage may prove useful in sensing this change and thereby allowing drawing conclusions on adsorption or desorption processes, complementing techniques such as surface plasmon resonance spectroscopy.

Fig. 6 Change of D upon variation of the dielectric functions of the two metals at an angle of incidence of θ ₀ = 80°.

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In summary, we have demonstrated in this paper the existence of the analogue of the Goos-Hänchen effect for SPPs at a gold-aluminum interface. Due to the large critical angle of θ _c ≃ 75° for total internal reflection of the SPP, lateral displacements of several times the wavelength of the incident beam occur. The sensitivity of the displacement to changes of the surface optical properties may be exploited to measure, for instance, the modification of the dielectric environment of the metal upon molecular adsorption.

Acknowledgments

BB acknowledges the MMM Initiative of the Max Planck Society. We thank D. Andrienko for a critical reading of the manuscript. FH is grateful to C. Sommer for helpful discussions. TAL and AAM acknowledge support from AFRL contract FA9453-08-C-0230.

References and links

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The Goos-Hänchen effect for surface plasmon polaritons

Abstract

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Acknowledgments

References and links

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