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Polarization-dependent Goos-Hänchen shift at a graded dielectric interface W. Löfflera , M. P. van Extera , G. W. ’t Hoofta , E. R. Eliela , K. Hermansb , D. J. Broerb , J. P. Woerdmana a Huygens Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands of Technology Eindhoven, PO Box 513, 5600 MB Eindhoven, The Netherlands b University Abstract We examine the polarization differential Goos-Hänchen beam shift upon total internal reflection, for a graded-index dielectric interface. We find a generic scaling law where the magnitude of this shift depends solely on the product of wavelength and gradient steepness. The analytic results are extended using transmission matrix calculations in cases where the assumptions made to allow analytical treatment might become questionable. Two important cases in this category are: (i) incident angle close to the critical angle and (ii) gradients with an overall thickness of the order of a wavelength. We demonstrate this effect experimentally using a polymer-blend sample with a gradual refractive-index transition induced by diffusion. Keywords: graded-index (GRIN) optics, Goos-Hänchen (GH) shift, polarization, inhomogeneous media 1. Introduction When a light beam undergoes total internal reflection 100% of its energy is reflected. Because evanescent fields penetrate into the lower-index medium, the reflected beam does not follow the geometrically expected path, but is translated a little, the amount depending on the polarization of the incident light. This translation, known as the Goos-Hänchen (GH) shift [1], is a correction to geometric optics. It arises because the planewave components of the incident beam pick up a phase jump upon reflection that depends on the incident angle. The GH effect has been studied for interfaces separating homogeneous dielectric media [1] and metamaterials [2], as well as for metallic [3] and absorbing [4, 5] layers. In inhomogeneous media, the polarizationdependent GH effect is unexplored yet; since abrupt interfaces are a special case of graded transitions, new insight can be obtained by comparing these two cases (See, e.g., [6, 7]). The characteristic geometric parameter of total internal reflection, i.e. the critical angle, is the same for gradual and abrupt transitions but the beam is not abruptly but gradually steered into the reflected direction. This brings us to the question that we address here: How does the (polarization-dependent) GH beam shift depend on the refractive index profile? Email address: [email protected] (W. Löffler) Preprint submitted to Optics Communications Since graded-index (GRIN) dielectrics are very important for waveguide optics, the influence of polarization on the phase of a propagating mode has been investigated theoretically in a cylindrical geometry [8]. However no analytical results or experimental tests for the polarization differential GH beam displacement in GRIN media have been reported. The GH shift in 1D GRIN media has only been studied theoretically, for perpendicular s (TE) polarization [9]. A related effect, the transverse shift of a circularly polarized beam in a GRIN medium can be described within the Berry-phase framework [10]. Recently, Bliokh et al. [11] developed an analytic theory describing ray paths in general inhomogeneous media emphasizing geometric phases. In principle the analytic part of our investigations is contained in this theory, we however find direct analytic and numeric results for the special, experimentally attractive case of a planar (1D) structure, which was not discussed before [10, 11]. Here we discuss only the case of total reflection, tunneling of light in GRIN media has been studied in [12, 13]. The 1D GRIN medium is also the simplest case in the very young field of transformation optics [14, 15]: There, the full potential of inhomogeneous media is demonstrated, concealing objects and allowing omnidirectional retro-reflection to name a few examples. The GH shift that we discuss here marks a general limit of such devices. In this paper we first give an analytical theory of the polarization-dependent GH beam shift for a graded inApril 21, 2010 [17] Hy (x) = n(x) g(x), which results in h i ∂2x g + n2 k02 − β2 + δV(x) g = 0 δV(x) = with (3) 2 1 2 ∂ n − 2 (∂ x n)2 . n x n (4) For sufficiently smooth GRIN media we can treat δV as a small perturbation. This allows us to approximate the propagation constant in the x-direction, k x,p , as Figure 1: (Color online) Scheme of the experiment: A light beam impinges from the high-refractive index (n1 ) material upon the gradedindex (GRIN) transition layer where the refractive index gradually reaches n2 . Right: Exemplary refractive index profile. The turning point of the beam is at x = xtp . k2x,p = k2x " #2 δV(x) + δV(x) ≈ k x + . 2k x (5) Our problem is ideally suited to be solved using the WKB method (see, e.g., [18]): We consider “smooth” index variations where the refractive index changes only little over a wavelength. The path of the beam is treated ray-optically, but we keep track of the phase. The acquired phase starting at x = 0 up to the turning point x = xtp and back (see Fig. 1) is for s polarization Z xtp φs = 2 k x (x0 )dx0 (6) terface, subsequently we extend the results using numerical simulation for cases where the assumptions made for analytical treatment are questionable. Finally, we give a first experimental demonstration of the effect. 2. Theory and numerical simulations 0 As a starting point of our analytic description, we take a linearly polarized Gaussian light beam that undergoes total internal reflection in a 1D graded dielectric structure (see Fig. 1). Because we are interested in beam displacements in the plane of incidence, it is sufficient to analyze Maxwell’s equation in two dimensions. For simplicity we assume an infinitely extended refractive-index gradient. The gradient is along the x axis and n(x) is decreasing with positive x; the whole system is translationally invariant in the y-direction. We assume n to be real, i.e. neglect absorption. The fields can be separated like E(r) = E(x) exp(iβz) for the electric field and H(r) = H(x) exp(iβz) for the magnetic field (β ≡ kz is the propagation constant parallel to the interface). We only need to consider one component of the fields E and H, Ey and Hy . For s polarization (TE, Ey (x) component) the wave equation is (see, e.g., Ref. [8]): (1) ∂2x Ey + n2 k02 − β2 Ey = 0 and for p polarization (TM, Hy (x) component) ∂2x Hy − ∂ x ln n2 ∂ x Hy + n2 k02 − β2 Hy = 0, while for p polarization: Z xtp Z φp = 2 k x,p (x0 )dx0 ≈ φ s + 0 0 xtp δV(x0 ) 0 dx (7) k x (x0 ) The absolute GH shift at a GRIN interface has little meaning since the reflection plane is undefined. However, the (interesting) physical differences of s and p polarized beams manifests themselves in the relative beam displacement; this is evident by comparing Eqs. 6 and 7. Therefore we focus here on the reflection phase difference between s and p polarization: Z xtp δV(x0 ) 0 φ p − φs = dx (8) k x (x0 ) 0 For easy interpretation we introduce a scaling parameter x0 , being a measure of the width of the gradient, and rewrite the refractive index profile as n(x) = n0 f (x/x0 ) where n0 is a constant. We introduce the incident angle as θinc , β ≡ o−1 n0 sin θinc with on0 ≡ λ/[2π n(0)], and x̃ ≡ x0 /x0 . Then the differential phase shift is (2) where ∂ x is the partial derivative with respect to the xcoordinate; the vacuum wave number is k0 = 2π/λ. We note that Eq. 2 is not analytically solvable for a general refractive index profile n(x) (see [16]). The TM (p) case can be rewritten into a form that resembles the TE (s) case by making the substitution φ p −φ s = on0 x0 Z 0 x̃tp f ( x̃) f 00 ( x̃) − 2 f 0 ( x̃) 2 d x̃ (9) q 2 2 2 f ( x̃) f ( x̃) − sin θinc This differential phase is dependent on on0 and therefore essentially different from the conventional phase shift at an abrupt interface, which is independent of 2 D ps = −on0 ∂ φ p − φs ∂θinc o2n0 . x0 Goos-Hänchen shift Dps (µm) 0 µm 3 1 µm 2 µm 3 µm 4 µm 1 2 Refractive-index profiles 1 .5 1 .4 4 1 .3 1 .2 1 .1 1 .0 0 1 2 3 4 Position (µm) 5 λ = 675 nm 0 4 1 .5 4 2 .0 4 2 .5 4 3 .0 4 3 .5 Incident angle (°) 4 4 .0 4 4 .5 4 5 .0 Figure 2: (Color online) Transmission matrix calculation of the differential GH shift D ps as a function of incident angle. Shown are calculations for different gradient thicknesses, the inset shows the profiles. (10) The first index profile we discuss is a linear refractive index gradient n( x̃) = n0 (1 − x̃) in the configuration shown in Fig. 1. The resulting differential GH shift is plotted in Fig. 2 for different gradient thicknesses. The largest shift (black curve) depicts the well-known case of an abrupt interface. We focus first on angles of incidence away from the critical angle. By introducing a gradual index change the differential GH shift is strongly reduced, already for a 1 µm thick gradient, and more so for 2, 3 and 4 µm thick gradients. This reflects the scaling law of Eq. 11: by increasing the gradient width the scaling factor on0 /x0 → 0. At the critical angle, the conventional differential GH shift (at a hard interface) diverges, this remains true for the GRIN case. However, for θ > θcrit , the shift is reduced more quickly for smoother interfaces than for abrupt ones (Fig. 2). Now we focus on the influence of the shape of the refractive-index transition on the differential GH beam shift. To this end we calculated the GH shift for different profiles: linear, complementary error function (profile achieved by many diffusion-driven processes [23]), and hyperbolic secant (ideal profile for graded-index fibers [8]). The refractive-index difference is kept constant (0.5) and the profiles have a comparable width of the index gradient (see inset Fig. 3). From an experimental point of view, the differential GH shift curves for different profiles show a similar reduction (compared to the abrupt interface case, black curve in Fig. 3); this is well explained by the overall scaling of the GH shift in GRIN media (Eq. 11). More precisely, the GH shift follows this scaling only on average; the details of the GH shift D ps (θinc ) are indeed Substituting Eq. 9 into Eq. 10 leads to the main result D ps ∼ Gradient thickness 5 Refractive index wavelength (i.e. zero-order in on0 ). The latter disappears in GRIN media and Eq. 9 gives the first-order correction. This was also found by Liberman and Zel’dovich [19] as well as by Bliokh and Stepanovskii [20]; they retrieved equivalent results for the phase difference. Our expression is more straightforward owing to the introduction of the scaling factor x0 . The integral factor in Eq. 9 depends on how smooth the index transition from f (0) = 1 to f ( x̃tp ) = sin θinc is; in general the integral is negative and of order O(1). For quadratic profiles of the form n2 (x) = n2o (1 − x2 /x02 ) the integral is approximately −π. In contrast to [19, 20] we are interested in the differential parallel beam displacement (D ps = D p − D s ). This is calculated from Artmann’s formula [21] and depends only on the derivative of the reflection phases ∂φ p,s /∂θinc : (11) This highlights the scaling of the differential GoosHänchen shift at a GRIN interface; it vanishes if the refractive-index transition region x0 is much larger than the (reduced) wavelength on0 . The conventional firstorder (in on0 ) differential GH shift disappears in a graded-index medium; from Eq. 11 we learn that only the second-order differential GH shift survives. The analytic solution (Eq. 9 and 10) has three limitations: (i) close to the critical angle the finite spatial extent of the index gradient becomes apparent, (ii) steep wavelength-scale gradients violate the slowly-varying potential requirement for the WKB method, and (iii) the role of the exact shape of the refractive-index profile remains to be quantified. These cases require numerical calculation. The transmission matrix method [22] is ideally suited for numerical calculation of the reflection phase for a 1D GRIN structure. The GRIN medium is approximated as a multilayer structure with 10 nm/layer (at λ0 = 675 nm). We verified that this is sufficiently finegrained. We obtain the reflection phase depending on incident angle, and using Artmann’s formula (Eq. 10) the differential GH displacement D ps of p and s polarized beams upon passage through the GRIN structure is determined. All calculations are for the glass (n = 1.5) – air (n = 1.0) transition. 3 erfc Refractive-index 1 .5 3 1 .4 Refractive index Goos-Hänchen shift Dps (µm) lin, 0 µm 4 tion. The system we have chosen is a graded polymerblend made out of ethoxylated Bisphenol-A diacrylate (EBADA) and tetrafluoropropylmethacrylate (TFPMA) with refractive indices of 1.54 and 1.42, respectively and negligible absorption in the relevant spectral range. To induce polymerization by UV light the monomers were mixed with the photo-initiator IRGACURE 819 (EBADA with 1 %, TFPMA with 2 % by weight). The demonstration sample was prepared directly on a face of an equilateral BK7 prism. Below a cover slip, an approximately 130 µm thick film of the EBADA was deposited and (partially) polymerized initiated by a short UV exposure, with a dose below that for full crosslinking. Using the same technique, a layer of TFPMA was deposited on top of this layer. During the diffusion time T D the system was kept undisturbed to allow diffusion of the monomer TFPMA into the partially cross-linked matrix of EBADA. Finally, the whole sample was fully polymerized by UV exposure. The refractive indices of the single polymer layers have been determined to be 1.54 (EBADA) and 1.42 (TFPMA). This corresponds to a critical angle of 67.23◦ , which also applies for the graded interface. Collimated light from a superluminescent diode (λ = 677 nm, ∆λ = 9 nm) was incident on the sample in a θ − 2θ setup. This light was modulated continuously between s and p polarization using a photo-elastic modulator. The differential positional shift of the reflected beams for s and p polarization was detected using a silicon quadrant photodiode as detector, together with a lock-in amplifier [26]; correction for Fresnel refraction at all interfaces has been done. A residual birefringence is present in the high-index layer, this results in an offset in the differential displacement signal. For clarity, we took care of this by vertically shifting the experimental curve. The resulting differential GH shift close to the critical angle is shown in Fig. 4: We see clear indication that the relative displacement is strongly reduced for a GRIN interface compared to the hard interface case. The theoretically expected GH shift for a 2 µm GRIN is plotted for comparison. profiles 5 1 .3 1 .2 sech 1 .1 lin, 2 µm 1 .0 2 0 .0 erfc 0 1 sech 4 2 .0 0 .5 1 .0 2 .0 λ = 675 nm lin, 2 µm 4 2 .5 1 .5 Position (µm) 4 3 .0 4 3 .5 4 4 .0 Incident angle (°) 4 4 .5 4 5 .0 Figure 3: (Color online) Transmission matrix calculations of the differential beam shift for different refractive-index profiles: linear, complementary error function, and hyperbolic secant (the black curve shows the case of an abrupt interface for comparison). The inset shows the GRIN profiles. Figure 4: Experimental GH shift for the polymer-blend GRIN sample in comparison with the theoretical curve for a 2 µm thick gradient (n = 1.54...1.42). The experimental curve is displaced vertically as explained in the text. The inset sketches the experimental configuration. highly sensitive to the GRIN profile shape. This is because WKB scattering amplitudes are very sensitive to the potential profile [24, 25]. 4. Conclusions In conclusion, smoothening of a refractive index gradient results in a reduction of the polarization differential Goos-Hänchen effect if the GRIN thickness is of the order of or larger than a wavelength. This can be seen as a manifestation of decoupling of polarization and spatial degrees of freedom in inhomogeneous media, provided 3. Experimental demonstration For a first experimental demonstration we require samples with a large refractive-index difference to (i) maximize the GH shift and to (ii) achieve an experimentally accessible angle of total internal reflec4 that the inhomogeneity on0 /x0 is small. The differential Goos-Hänchen effect in GRIN media is a second-order correction to geometric optics; the well-known firstorder differential Goos-Hänchen shift disappears here. This generalizes the conventional Goos-Hänchen effect. [21] K. Artmann, Berechnung der Seitenversetzung des totalreflektierten Strahles, Ann. Phys. 437 (1948) 87–102. [22] R. M. Azzam, N. M. 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