Abstract

We present a theoretical study of the Goos-Hänchen and Imbert-Fedorov shifts for a fundamental Gaussian beam impinging on a surface coated with a single layer of graphene. We show that the graphene surface conductivity σ(ω) is responsible for the appearance of a giant and negative spatial Goos-Hänchen shift.

© 2015 Optical Society of America

1. Introduction

When an optical beam impinges upon a surface, nonspecular reflection phenomena may occur, such as the Goos-Hänchen (GH) [1–3] and Imbert-Fedorov (IF) [4, 5] shifts, resulting in an effective beam shift at the interface. A comprehensive review on beam shift phenomena can be found in Ref. [6]. Although Goos and Hänchen published their work more than 60 years ago [1], this field of research is still very active, and in the last decades a vast amount of literature has been produced on the subject, resulting not only in a better understanding of the underlying physical principles [7–14], but also in the careful investigation of the effects of various field configurations [15–19] and reflecting surfaces [20–23] on the GH and IF shifts. In the last decade, moreover, the giant GH shift has also been observed to occur in various systems, such as metamaterials [24], photonic crystals [25] and complex crystals [26].

In recent years, on a parallel trail, graphene attracted very rapidly a lot of interest, thanks to its intriguing properties [27, 28]. Its peculiar band structure and the existence of the so-called Dirac cones [29], for example, gives the possibility to use graphene as a model to observe QED-like effects such as Klein tunneling [30], Zitterbewegung [31], the anomalous quantum Hall effect [32] and the appearance of a minimal conductivity that approaches the quantum limit e2 for vanishing charge density [33]. In addition, the reflectance and transmittance of graphene are determined by the fine structure constant [34], and a single layer of graphene shows universal absorbance in the spectral range from near-infrared to the visible part of the spectrum [35].

Among the vast plethora of applications, graphene also proved to be a very interesting system where to observe beam shifts. Very recently, in fact, the occurrence of GH shift in graphene-based structures has been reported, both for light beams (where giant GH shift has been observed [36]) and for Dirac fermions [37]. Contextually, the occurrence of giant GH shift for a graphene layer embedded in two media with different permittivity has been predicted by Martinez and co-workers in 2001 [38]. Despite this theoretical work, however, a full theoretical analysis of both spatial and angular GH and IF shifts has not been carried out yet.

In this work, we therefore present a theoretical analysis of the GH and IF shifts occurring for a monochromatic Gaussian beam impinging onto a glass surface coated with a single layer of graphene. The results of our investigations show on one hand, that the appearance of a giant GH shift is ultimately due to the graphene’s surface conductivity σ(ω) (in accordance with the results of Ref. [38]), and on the other hand, that the presence of the single layer of graphene introduces a dependence of the phases of the reflection coefficients on the incidence angle, thus resulting in a nonzero spatial GH shift also when total internal reflection does not occur. Our results are in agreement with previous investigations of beam shifts in lossy surfaces [39] or multilayered media [17], where the appearance of a nonzero spatial shift due to the complexvaluedness of the reflection coefficient resulting from the nontrivial structure of the surface has also been reported.

We start our analysis by considering a monochromatic Gaussian beam with frequency ω = ck0 (with k0 being the vacuum wave number), impinging on a dielectric surface characterized by the refractive index n and coated with a single layer of graphene [Fig. 1(a)]. The graphene layer is characterized by the optical conductivity σ(ω), whose expression can be given in the following dimensionless form [29]

σ(Ω)σ0=Θ(Ω2)+i[1πΩ1πln|Ω+2Ω2|],
where σ0 = e2/(4ħ) = παcε0 (being α ≈ 1/137 the fine structure constant [31]) is the universal optical conductivity of graphene, Ω = ħω/µ is the dimensionless frequency, µ is the chemical potential and Θ(x) is the Heaviside step function [40]. Later, we use Ω = 2.5 as an exemplary value for all figures.

 figure: Fig. 1

Fig. 1 (a) Schematic representation of the considered surface. The graphene layer (green) is characterized by its surface conductivity σ(Ω), whose explicit expression is given by Eq. (1). The dielectric substrate (green) is characterized by the refractive index n. (b) Geometry of beam reflection at the interface. The single graphene layer is located on the surface at z = 0. The different Cartesian coordinate systems K,Ki,Kr are shown.

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According to Fig. 1(b), we define three Cartesian reference frames: the laboratory frame K = (O,x,y,z) attached to the reflecting surface, the (local) incident frame Ki = (O,xi,y,zi) attached to the incident beam, and the (local) reflected frame Kr = (O,xr,y,zr) attached to the reflected beam. These three reference frames are connected via a rotation of an angle θ around the y direction [39]. The reflecting surface is located at z = 0, with the z-axis pointing towards the interface. With this choice of geometry, the incident beam comes from the region z < 0 and propagates in the x-z plane.

The electric field in the incident frame can be then written, using its angular spectrum representation [41], as follows:

Ei(r)=l=12d2Ke^λ(U,V,θ)Aλ(U,V,θ)eikiri,
where d2K = dUdV, êλ (U,V,θ) is the local reference frame attached to the incident field [42], Aλ (U,V,θ) = αλ (U,V,θ)A(U,V) and ki · ri = UXi +VYi +WZi, being Xi = k0xi the normalized coordinate in the incident frame. Yi and Zi are defined in a similar manner. αλ(U,V,θ)=e^λ(U,V,θ)f^ accounts for the projection of the beam’s polarization f^=fpx^+fsy^ (normalized according to |fp|2+|fs|2=1) onto the local basis, and A(U,V) is the beam’s spectral amplitude, which here is assumed to be Gaussian, i.e.,
A(U,V)=ew02(U2+V2),
being w02 the spot size of the beam. In the remaining of the manuscript, we will consider only well collimated beams, namely the paraxial assumption U,V ≪ 1 is implicitly understood.

Upon reflection, the electric field can be then written as follows:

Ei(rr)=λ=12d2Ke^λ(U,V,πθ)A˜λ(U,V,θ)eikrrr,
where kr · rr = −UXr +VYr +WZr and Ãλ (U,V,θ) = rλ (U,V,θ)Aλ (U,V,θ), with rλ (U,V,θ) being the Fresnel reflection coefficients associated to the single plane wave component of the field [43]. The minus sign in front of U in êλ, as well as in kr · rr accounts for the specular reflection of the single plane wave component [42].

The presence of a single layer of graphene deposited on the dielectric surface modifies its reflection coefficients as follows [44]:

rs(θ)=cosθn2sin2θσ(Ω)cosθ+n2sin2θ+σ(Ω),
rp(θ)=n2cosθn2sin2θ[1σ(Ω)cosθ]n2cosθ+n2sin2θ[1+σ(Ω)cosθ],
where θ is the incident angle, n is the refractive index of the dielectric medium and σ(Ω) is the graphene’s surface conductivity, as defined by Eq. (1). The modulus Rλ and phase ϕλ of the reflection coefficients rλ=Rλeiϕλ (with λ ∈ {p,s}) are shown in Fig. 2, together with the correspondent quantities for the case of a simple dielectric surface without the graphene coating. While the presence of the graphene layer does not modify significatively Rλ for neither p- or s-polarization (as it appears clear from Figs. 2(a) and (c), respectively), the change induced in the phases ϕλ of the reflection coefficients is considerable. For an air-glass interface, in fact, we have ∂ϕλ/∂θ = 0 being θ the angle of incidence. Here, instead, we have ∂ϕλ/∂θ ≠ 0. A closer inspection of Eqs. (5), moreover, reveals that such a novel θ-dependence of the phases ϕλ is entirely due to the graphene conductivity σ(Ω).

 figure: Fig. 2

Fig. 2 Modulus (left column) and phase (right column) of the reflection coefficients rλ = Rλ exp(λ) for p-polarization (top row) and s-polarization (bottom row). In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the bulk medium is chosen to be n = 1.5.

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To compute the GH and IF shifts, we calculate the center of mass of the intensity distribution in the reflected frame, namely [42]

R=+R|Er|2dXrdYr+|Er|2dXrdYr=XrX^r+Yry^r,
where R = (Xr,Yr)T. Spatial (Δ) and angular (Θ) GH and IF shifts are then defined as follows:
k0ΔGH=Xr|z=0,ΘGH=Xrz,
k0ΔIF=Yr|z=0,ΘIF=Yrz.

The explicit expressions of the GH and IF shifts for a fundamental Gaussian beam read, according to [45], as follows:

k0ΔGH=wpϕpθ+wsϕsθ,
k0ΔIF=cotθ[wpas2+wsap2apassinη+2wpwssin(ηϕp+ϕs)],
ΘGH=(wplnRpθ+wslnRsθ),
ΘIF=wpas2wsap1apascosηcotθ,
where fp = ap, fs = as exp() and wλ=aλ2Rλ2/(ap2Rp2+as2Rs2) (where λ ∈ {p,s}) is the fractional energy contained in each polarization state.

As suggested by Figs. 2(a) and (c), the changes in Rλ introduced by the graphene layer are negligible. We therefore expect to observe no changes in the angular shifts ΘGH and ΘIF, as they are functions of Rλ solely. The spatial shifts ΔGH and ΔIF, on the other hand, contain a dependence on the phases ϕλ, and they are therefore affected by the presence of the graphene coating. Let us first discuss the IF shift. In this case ϕpϕs is very close to π [Figs. 2(b) and (d)], and the resulting spatial shift ΔIF will be nonzero (but very small) even for linear polarization, in contrast to the case without graphene. The adimensional spatial IF shift is depicted in Fig. 3(a) and (b) for circular and linear polarized light, respectively.

 figure: Fig. 3

Fig. 3 Spatial IF shift ΔIF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface. The spatial IF shifts are nonzero even for linear polarization. Spatial GH shift ΔGH for (c) p- polarization and (d) s-polarization for a graphene-coated surface. Since ∂ϕλ/∂θ ≠ 0, in both cases ΔGH ≠ 0. In particular, since ϕp varies very rapidly with θ in the vicinity of the Brewster angle, the corresponding spatial shift for p-polarization [Panel (c)] is giant in modulus, and negative due to the fact that ϕp varies from 0 to −π [See Fig. 2(b)].

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More interesting is the case of the spatial GH shift. For an air-dielectric interface, one has ∂ϕλ/∂θ = 0 and therefore, according to Eq. (8a), ΔGH = 0. It is in fact well known since the pioneering work of Goos and Hänchen [1], that ΔGH = 0 occurs only in total internal reflection, where Rλ = 1 and ∂ϕλ/∂θ = 0. For the case of a graphene-coated surface, on the other hand, the phase ϕλ varies with θ for both s- and p-polarizations, as Figs. 2(b) and (d), respectively, show. In this case, then, we observe a nonzero spatial GH shift even without total internal reflection.

The adimensional spatial GH shift k0ΔGH occurring at a graphene-coated dielectric surface is depicted in Fig. 3(c) and (d) for p- and s-polarization, respectively. As can be seen, for both polarizations we have ΔGH ≠ 0 although no total internal reflection takes place. In particular, ϕp varies very rapidly from 0 to −π in the vicinity of the Brewster angle θB. This corresponds to a giant and negative spatial GH shift. On the other hand, ϕs varies very smoothly with θ, thus resulting in a nonzero (but very small) spatial GH shift for s-polarization.

It is also instructive to calculate the GH shift for the case of total internal reflection (TIR). In this case, the reflection coefficients given by Eqs. (5) need to be modified to account for the fact that now the reflection takes place from the dielectric medium to air (where TIR can actually occur). Following the results of Ref. [44], the reflection coefficients for s–and p−polarization are then given by

rs(θ)=ncosθ1n2sin2θnσ(Ω)ncosθ+1n2sin2θ+nσ(Ω),
rp(θ)=cosθn1n2sin2θ[1σ(Ω)cosθ]ncosθ+n1n2sin2θ[1+σ(Ω)cosθ],
where θ is the incident angle, n is the refractive index of the dielectric medium and σ(Ω) is the graphene’s surface conductivity, as defined by Eq. (1). The reflection coefficients for internal reflection are depicted in Fig. 4.

 figure: Fig. 4

Fig. 4 Modulus (left column) and phase (right column) of the reflection coefficients rλ = Rλ exp(λ) for p-polarization (top row) and s-polarization (bottom row) for the case with TIR. In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the dielectric medium is chosen to be n = 1.5.

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The calculation of the GH and IF shifts for this case are essentially the same as explained above, with the only difference that upon TIR the reflection coefficients are close to 1 and vary slowly. According to Eqs. (8), this results in having θGH 0, θIF 0 for the angular and

k0ΔGH=wpϕpθ+wsϕsθ,
k0ΔIF=4apasap2+as2cotθsin(ηϕpϕs2)cos(ϕpϕs2),
for the spatial shifts.

The adimensional spatial GH and IF shifts for TIR are depicted in Fig. 5. Again, the spatial shifts ΔGH and ΔIF contain a dependence on the phases ϕλ, and they are therefore affected by the presence of the graphene coating. The spatial IF is nonzero for both circular polarized light and for 45° linear polarized light. Above the critical angle the dependence is altered compared to the case without graphene and a negative shifts occurs even for linear polarized light. For the spatial GH shift, a giant negative shift is observed near the critical angle of the TIR for both, sand p-polarized light. Again, these shifts can be understood as a variation of the phases ϕλ.

 figure: Fig. 5

Fig. 5 Spatial IF shift ΔIF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface for the case with TIR. Spatial GH shift ΔGH for (c) p- polarization and (d) s-polarization for a graphene-coated surface for the case with TIR.

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2. Conclusion

In conclusion, we have presented a detailed theoretical analysis of GH and IF shifts of a Gaussian beam impinging onto a graphene-coated dielectric surface. Our analysis revealed that the main effect of the graphene layer is to introduce, through its surface conductivity σ(ω), a dependence of the phases ϕλ of the reflection coefficients on the incident angle θ. This ultimately reflects in the appearance of a nonzero spatial GH and IF shifts. In particular a giant and negative spatial GH shift in the vicinity of the Brewster’s angle for p-polarization and additionally in the case with TIR at the critical angle for p- and s-polarization has been predicted. For the case of TIR, our calculations proved to be in agreement with the recently published experimental results [36].

Acknowledgments

The authors thank the German Ministry of Education and Science ( ZIK 03Z1HN31) for financial support.

References and links

1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333 (1947). [CrossRef]  

2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 2, 87 (1948). [CrossRef]  

3. K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen Effect: A Simple Example of a Time Delay Scattering Process,” Am. J. Phys. 40, 1847 (1972) [CrossRef]  

4. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465 (1955).

5. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787 (1972). [CrossRef]  

6. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15, 014001 (2013). [CrossRef]  

7. F. Pillon, H. Gilles, and S. Girard, “Experimental observation of the Imbert-Fedorov transverse displacement after a single total reflection,” Appl. Opt. 43, 1863 (2004). [CrossRef]   [PubMed]  

8. H. Schilling, “Die Strahlversetzung bei der Reflexion linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. (Berlin) 16, 122 (1965). [CrossRef]  

9. M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. A: Math. Gen. 20, 3667 (1987). [CrossRef]  

10. V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. A: Math. Gen 21, 2045 (1988). [CrossRef]  

11. V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992). [CrossRef]   [PubMed]  

12. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006). [CrossRef]  

13. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437 (2008). [CrossRef]   [PubMed]  

14. O. Hosten and P. Kwiat, “Observation of the spin-hall effect of light via weak measurements,” Science 319, 787 (2008). [CrossRef]   [PubMed]  

15. P. T. Leung, C. W. Chen, and H. -P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206 (2007). [CrossRef]  

16. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. von Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928 (2007). [CrossRef]   [PubMed]  

17. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558 (1986). [CrossRef]  

18. G. D. Landry and T. A. Maldonado, “Gaussian beam trasmission and reflection from a general anisotropic multilayered structure,” Appl. Opt. 35, 5870 (1996). [CrossRef]   [PubMed]  

19. M. Ornigotti, A. Aiello, and C. Conti, “Goos-Hänchen and Imbert-Fedorov shifts for paraxial X-Waves,” Opt. Lett. 40, 558 (2015). [CrossRef]   [PubMed]  

20. S. Kozaki and H. Sakurai, “Characteristic of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508 (1978). [CrossRef]  

21. D. Golla and S. Dutta Gupta, “Goos-Hänchen shift for higher order Hermite-Gaussian beams,” arXiv:1011.3968v1.

22. M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010). [CrossRef]  

23. A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010). [CrossRef]  

24. I. V. Shadrivov, A. A. Zharov, and Yu S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Whys. Let. 83, 2713 (2003). [CrossRef]  

25. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shift for a photonic crystal with a negative effective index,” Opt. Express 14, 3024 (2006). [CrossRef]   [PubMed]  

26. S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011). [CrossRef]  

27. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005). [CrossRef]   [PubMed]  

28. A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009). [CrossRef]  

29. M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University Press, 2012). [CrossRef]  

30. A. Calogeracos and N. Dombey, “History and physics of the Klein paradox,” Contemp. Phys. 40, 313 (1999). [CrossRef]  

31. C. Itzykson and J. B. Zuber, Quantum Field Theory (Dover, 2006).

32. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005). [CrossRef]   [PubMed]  

33. M. I. Katsnelson, “Zitterbewegung, chirality, and minimal conductivity in graphene,” Eur. Phys. J. B. 51, 157 (2006). [CrossRef]  

34. R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008). [CrossRef]   [PubMed]  

35. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008). [CrossRef]  

36. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39, 5574 (2014). [CrossRef]   [PubMed]  

37. A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014). [CrossRef]  

38. J. C. Martinez and M. B. A. Janil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011). [CrossRef]  

39. M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009). [CrossRef]  

40. Digital Library of Mathematical Functions, http://dlmf.nist.gov, National Institute of Standard and Technology (2010).

41. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995). [CrossRef]  

42. A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near brewster incidence,” arXiv:0903.3730v2 [physics.optics].

43. M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, 2003).

44. T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).

45. M. Ornigotti and A. Aiello, “Goos-Hänchen and Imbert-Fedorov for bounded wave packets of light,” J. Opt. 15, 014004 (2013). [CrossRef]  

References

  • View by:

  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333 (1947).
    [Crossref]
  2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 2, 87 (1948).
    [Crossref]
  3. K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen Effect: A Simple Example of a Time Delay Scattering Process,” Am. J. Phys. 40, 1847 (1972)
    [Crossref]
  4. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465 (1955).
  5. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787 (1972).
    [Crossref]
  6. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15, 014001 (2013).
    [Crossref]
  7. F. Pillon, H. Gilles, and S. Girard, “Experimental observation of the Imbert-Fedorov transverse displacement after a single total reflection,” Appl. Opt. 43, 1863 (2004).
    [Crossref] [PubMed]
  8. H. Schilling, “Die Strahlversetzung bei der Reflexion linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. (Berlin) 16, 122 (1965).
    [Crossref]
  9. M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. A: Math. Gen. 20, 3667 (1987).
    [Crossref]
  10. V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. A: Math. Gen 21, 2045 (1988).
    [Crossref]
  11. V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992).
    [Crossref] [PubMed]
  12. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
    [Crossref]
  13. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437 (2008).
    [Crossref] [PubMed]
  14. O. Hosten and P. Kwiat, “Observation of the spin-hall effect of light via weak measurements,” Science 319, 787 (2008).
    [Crossref] [PubMed]
  15. P. T. Leung, C. W. Chen, and H. -P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206 (2007).
    [Crossref]
  16. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. von Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928 (2007).
    [Crossref] [PubMed]
  17. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558 (1986).
    [Crossref]
  18. G. D. Landry and T. A. Maldonado, “Gaussian beam trasmission and reflection from a general anisotropic multilayered structure,” Appl. Opt. 35, 5870 (1996).
    [Crossref] [PubMed]
  19. M. Ornigotti, A. Aiello, and C. Conti, “Goos-Hänchen and Imbert-Fedorov shifts for paraxial X-Waves,” Opt. Lett. 40, 558 (2015).
    [Crossref] [PubMed]
  20. S. Kozaki and H. Sakurai, “Characteristic of a Gaussian beam at a dielectric interface,” J. Opt. Soc. Am. 68, 508 (1978).
    [Crossref]
  21. D. Golla and S. Dutta Gupta, “Goos-Hänchen shift for higher order Hermite-Gaussian beams,” arXiv:1011.3968v1.
  22. M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
    [Crossref]
  23. A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010).
    [Crossref]
  24. I. V. Shadrivov, A. A. Zharov, and Yu S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Whys. Let. 83, 2713 (2003).
    [Crossref]
  25. J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shift for a photonic crystal with a negative effective index,” Opt. Express 14, 3024 (2006).
    [Crossref] [PubMed]
  26. S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011).
    [Crossref]
  27. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
    [Crossref] [PubMed]
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  32. Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
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    [Crossref] [PubMed]
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    [Crossref]
  36. X. Li, P. Wang, F. Xing, X. D. Chen, Z. B. Liu, and J. G. Tian, “Experimental observation of a giant Goos-Hänchen shift in graphene using a beam splitter scanning method,” Opt. Lett. 39, 5574 (2014).
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  37. A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014).
    [Crossref]
  38. J. C. Martinez and M. B. A. Janil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
    [Crossref]
  39. M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
    [Crossref]
  40. Digital Library of Mathematical Functions, http://dlmf.nist.gov , National Institute of Standard and Technology (2010).
  41. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995).
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  43. M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, 2003).
  44. T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).
  45. M. Ornigotti and A. Aiello, “Goos-Hänchen and Imbert-Fedorov for bounded wave packets of light,” J. Opt. 15, 014004 (2013).
    [Crossref]

2015 (1)

2014 (2)

2013 (3)

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15, 014001 (2013).
[Crossref]

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).

M. Ornigotti and A. Aiello, “Goos-Hänchen and Imbert-Fedorov for bounded wave packets of light,” J. Opt. 15, 014004 (2013).
[Crossref]

2011 (2)

J. C. Martinez and M. B. A. Janil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
[Crossref]

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011).
[Crossref]

2010 (2)

M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[Crossref]

A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010).
[Crossref]

2009 (2)

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
[Crossref]

2008 (4)

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437 (2008).
[Crossref] [PubMed]

O. Hosten and P. Kwiat, “Observation of the spin-hall effect of light via weak measurements,” Science 319, 787 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (3)

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref]

J. He, J. Yi, and S. He, “Giant negative Goos-Hänchen shift for a photonic crystal with a negative effective index,” Opt. Express 14, 3024 (2006).
[Crossref] [PubMed]

M. I. Katsnelson, “Zitterbewegung, chirality, and minimal conductivity in graphene,” Eur. Phys. J. B. 51, 157 (2006).
[Crossref]

2005 (2)

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

2004 (1)

2003 (1)

I. V. Shadrivov, A. A. Zharov, and Yu S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Whys. Let. 83, 2713 (2003).
[Crossref]

1999 (1)

A. Calogeracos and N. Dombey, “History and physics of the Klein paradox,” Contemp. Phys. 40, 313 (1999).
[Crossref]

1996 (1)

1992 (1)

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

1988 (1)

V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. A: Math. Gen 21, 2045 (1988).
[Crossref]

1987 (1)

M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. A: Math. Gen. 20, 3667 (1987).
[Crossref]

1986 (1)

1978 (1)

1972 (2)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787 (1972).
[Crossref]

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen Effect: A Simple Example of a Time Delay Scattering Process,” Am. J. Phys. 40, 1847 (1972)
[Crossref]

1965 (1)

H. Schilling, “Die Strahlversetzung bei der Reflexion linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. (Berlin) 16, 122 (1965).
[Crossref]

1955 (1)

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465 (1955).

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 2, 87 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333 (1947).
[Crossref]

’t Hooft, G. W.

Aiello, A.

M. Ornigotti, A. Aiello, and C. Conti, “Goos-Hänchen and Imbert-Fedorov shifts for paraxial X-Waves,” Opt. Lett. 40, 558 (2015).
[Crossref] [PubMed]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15, 014001 (2013).
[Crossref]

M. Ornigotti and A. Aiello, “Goos-Hänchen and Imbert-Fedorov for bounded wave packets of light,” J. Opt. 15, 014004 (2013).
[Crossref]

M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[Crossref]

A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010).
[Crossref]

M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
[Crossref]

A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437 (2008).
[Crossref] [PubMed]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. von Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928 (2007).
[Crossref] [PubMed]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near brewster incidence,” arXiv:0903.3730v2 [physics.optics].

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Phys. 2, 87 (1948).
[Crossref]

Bahloulia, H.

A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014).
[Crossref]

Blake, P.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overwiev,” J. Opt. 15, 014001 (2013).
[Crossref]

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref]

Bliokh, Y. P.

K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref]

Booth, T. J.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, 2003).

Calogeracos, A.

A. Calogeracos and N. Dombey, “History and physics of the Klein paradox,” Contemp. Phys. 40, 313 (1999).
[Crossref]

CastroNeto, A. H.

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Chen, C. W.

P. T. Leung, C. W. Chen, and H. -P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206 (2007).
[Crossref]

Chen, X. D.

Chiang, H. -P.

P. T. Leung, C. W. Chen, and H. -P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206 (2007).
[Crossref]

Chiu, K. W.

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen Effect: A Simple Example of a Time Delay Scattering Process,” Am. J. Phys. 40, 1847 (1972)
[Crossref]

Conti, C.

Dai, Y.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).

Della Valle, G.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011).
[Crossref]

Dombey, N.

A. Calogeracos and N. Dombey, “History and physics of the Klein paradox,” Contemp. Phys. 40, 313 (1999).
[Crossref]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Dutta Gupta, S.

D. Golla and S. Dutta Gupta, “Goos-Hänchen shift for higher order Hermite-Gaussian beams,” arXiv:1011.3968v1.

Eliel, E. R.

Fedorov, F. I.

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465 (1955).

Fedoseyev, V. G.

V. G. Fedoseyev, “Conservation laws and transverse motion of energy on reflection and transmission of electromagnetic waves,” J. Phys. A: Math. Gen 21, 2045 (1988).
[Crossref]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Geim, A. K.

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Gilles, H.

Girard, S.

Golla, D.

D. Golla and S. Dutta Gupta, “Goos-Hänchen shift for higher order Hermite-Gaussian beams,” arXiv:1011.3968v1.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333 (1947).
[Crossref]

Grigorenko, A. N.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Guinea, F.

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333 (1947).
[Crossref]

He, J.

He, S.

Hermosa, N.

M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[Crossref]

Hosten, O.

O. Hosten and P. Kwiat, “Observation of the spin-hall effect of light via weak measurements,” Science 319, 787 (2008).
[Crossref] [PubMed]

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787 (1972).
[Crossref]

Itzykson, C.

C. Itzykson and J. B. Zuber, Quantum Field Theory (Dover, 2006).

Janil, M. B. A.

J. C. Martinez and M. B. A. Janil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
[Crossref]

Jellala, A.

A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014).
[Crossref]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Katsnelson, M. I.

M. I. Katsnelson, “Zitterbewegung, chirality, and minimal conductivity in graphene,” Eur. Phys. J. B. 51, 157 (2006).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University Press, 2012).
[Crossref]

Kim, P.

Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
[Crossref] [PubMed]

Kivshar, Yu S.

I. V. Shadrivov, A. A. Zharov, and Yu S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Whys. Let. 83, 2713 (2003).
[Crossref]

Kozaki, S.

Kwiat, P.

O. Hosten and P. Kwiat, “Observation of the spin-hall effect of light via weak measurements,” Science 319, 787 (2008).
[Crossref] [PubMed]

Landry, G. D.

Leung, P. T.

P. T. Leung, C. W. Chen, and H. -P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206 (2007).
[Crossref]

Li, X.

Liberman, V. S.

V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

Liu, X.

T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).

Liu, Z. B.

Longhi, S.

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011).
[Crossref]

Maldonado, T. A.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995).
[Crossref]

Martinez, J. C.

J. C. Martinez and M. B. A. Janil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
[Crossref]

Merano, M.

M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[Crossref]

M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
[Crossref]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. von Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928 (2007).
[Crossref] [PubMed]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Nair, R. R.

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Novoselov, K. S.

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
[Crossref] [PubMed]

Ornigotti, M.

M. Ornigotti, A. Aiello, and C. Conti, “Goos-Hänchen and Imbert-Fedorov shifts for paraxial X-Waves,” Opt. Lett. 40, 558 (2015).
[Crossref] [PubMed]

M. Ornigotti and A. Aiello, “Goos-Hänchen and Imbert-Fedorov for bounded wave packets of light,” J. Opt. 15, 014004 (2013).
[Crossref]

Peres, N. M. R.

A. H. CastroNeto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
[Crossref]

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008).
[Crossref]

R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
[Crossref] [PubMed]

Pillon, F.

Player, M. A.

M. A. Player, “Angular momentum balance and transverse shift on reflection of light,” J. Phys. A: Math. Gen. 20, 3667 (1987).
[Crossref]

Quinn, J. J.

K. W. Chiu and J. J. Quinn, “On the Goos-Hänchen Effect: A Simple Example of a Time Delay Scattering Process,” Am. J. Phys. 40, 1847 (1972)
[Crossref]

Redouanic, I.

A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014).
[Crossref]

Sakurai, H.

Schilling, H.

H. Schilling, “Die Strahlversetzung bei der Reflexion linear oder elliptisch polarisierter ebener Wellen an der Trennebene zwischen absorbierenden Medien,” Ann. Phys. (Berlin) 16, 122 (1965).
[Crossref]

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R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim, “Fine Structure Constant Defines Visual Transparency of Graphene,” Science 320, 1308 (2008).
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Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
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Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
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A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010).
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M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
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[Crossref]

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T. Zhan, X. Shi, Y. Dai, X. Liu, and J. Zi, “Transfer matrix method for optics in graphene layers,” J. Phys.: Conden. Matt. 25, 215301 (2013).

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Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201 (2005).
[Crossref] [PubMed]

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I. V. Shadrivov, A. A. Zharov, and Yu S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Whys. Let. 83, 2713 (2003).
[Crossref]

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Appl. Opt. (2)

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Nature (2)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197 (2005).
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A. Aiello and J. P. Woerdman, “Goos-Hänchen and Imbert-Fedorov shifts of a non diffracting Bessel beam,” Opt. Lett 36, 543 (2010).
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Opt. Lett. (3)

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V. S. Liberman and B. Y. Zel’dovich, “Spin-orbit interaction of a photon in an inhomogeneous medium,” Phys. Rev. A 46, 5199 (1992).
[Crossref] [PubMed]

S. Longhi, G. Della Valle, and K. Staliunas, “Goos-Hänchen shift in complex crystals,” Phys. Rev. A 84, 042119 (2011).
[Crossref]

M. Merano, N. Hermosa, J.P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82, 023817 (2010).
[Crossref]

M. Merano, A. Aiello, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80, 061801 (2009).
[Crossref]

Phys. Rev. B (1)

T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78, 085432 (2008).
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K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
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Physica E (1)

A. Jellala, I. Redouanic, Y. Zahidic, and H. Bahloulia, “Goos-Hänchen like shifts in graphene double barriers,” Physica E 58, 30 (2014).
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C. Itzykson and J. B. Zuber, Quantum Field Theory (Dover, 2006).

M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University Press, 2012).
[Crossref]

D. Golla and S. Dutta Gupta, “Goos-Hänchen shift for higher order Hermite-Gaussian beams,” arXiv:1011.3968v1.

Digital Library of Mathematical Functions, http://dlmf.nist.gov , National Institute of Standard and Technology (2010).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995).
[Crossref]

A. Aiello and J. P. Woerdman, “Theory of angular Goos-Hänchen shift near brewster incidence,” arXiv:0903.3730v2 [physics.optics].

M. Born and E. Wolf, Principles of Optics, 7th edition (Cambridge University Press, 2003).

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic representation of the considered surface. The graphene layer (green) is characterized by its surface conductivity σ(Ω), whose explicit expression is given by Eq. (1). The dielectric substrate (green) is characterized by the refractive index n. (b) Geometry of beam reflection at the interface. The single graphene layer is located on the surface at z = 0. The different Cartesian coordinate systems K,Ki,Kr are shown.
Fig. 2
Fig. 2 Modulus (left column) and phase (right column) of the reflection coefficients rλ = Rλ exp(λ) for p-polarization (top row) and s-polarization (bottom row). In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the bulk medium is chosen to be n = 1.5.
Fig. 3
Fig. 3 Spatial IF shift ΔIF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface. The spatial IF shifts are nonzero even for linear polarization. Spatial GH shift ΔGH for (c) p- polarization and (d) s-polarization for a graphene-coated surface. Since ∂ϕλ/∂θ ≠ 0, in both cases ΔGH ≠ 0. In particular, since ϕp varies very rapidly with θ in the vicinity of the Brewster angle, the corresponding spatial shift for p-polarization [Panel (c)] is giant in modulus, and negative due to the fact that ϕp varies from 0 to −π [See Fig. 2(b)].
Fig. 4
Fig. 4 Modulus (left column) and phase (right column) of the reflection coefficients rλ = Rλ exp(λ) for p-polarization (top row) and s-polarization (bottom row) for the case with TIR. In all graphs, the solid black line corresponds to the case of the graphene-coated surface, while the green dashed curve corresponds to the case without graphene coating. The refractive index of the dielectric medium is chosen to be n = 1.5.
Fig. 5
Fig. 5 Spatial IF shift ΔIF for (a) circular polarization and (b) 45° linear polarization for a graphene coated surface for the case with TIR. Spatial GH shift ΔGH for (c) p- polarization and (d) s-polarization for a graphene-coated surface for the case with TIR.

Equations (17)

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σ ( Ω ) σ 0 = Θ ( Ω 2 ) + i [ 1 π Ω 1 π ln | Ω + 2 Ω 2 | ] ,
E i ( r ) = l = 1 2 d 2 K e ^ λ ( U , V , θ ) A λ ( U , V , θ ) e i k i r i ,
A ( U , V ) = e w 0 2 ( U 2 + V 2 ) ,
E i ( r r ) = λ = 1 2 d 2 K e ^ λ ( U , V , π θ ) A ˜ λ ( U , V , θ ) e i k r r r ,
r s ( θ ) = cos θ n 2 sin 2 θ σ ( Ω ) cos θ + n 2 sin 2 θ + σ ( Ω ) ,
r p ( θ ) = n 2 cos θ n 2 sin 2 θ [ 1 σ ( Ω ) cos θ ] n 2 cos θ + n 2 sin 2 θ [ 1 + σ ( Ω ) cos θ ] ,
R = + R | E r | 2 d X r d Y r + | E r | 2 d X r d Y r = X r X ^ r + Y r y ^ r ,
k 0 Δ GH = X r | z = 0 , Θ GH = X r z ,
k 0 Δ IF = Y r | z = 0 , Θ IF = Y r z .
k 0 Δ GH = w p ϕ p θ + w s ϕ s θ ,
k 0 Δ IF = cot θ [ w p a s 2 + w s a p 2 a p a s sin η + 2 w p w s sin ( η ϕ p + ϕ s ) ] ,
Θ GH = ( w p ln R p θ + w s ln R s θ ) ,
Θ IF = w p a s 2 w s a p 1 a p a s cos η cot θ ,
r s ( θ ) = n cos θ 1 n 2 sin 2 θ n σ ( Ω ) n cos θ + 1 n 2 sin 2 θ + n σ ( Ω ) ,
r p ( θ ) = cos θ n 1 n 2 sin 2 θ [ 1 σ ( Ω ) cos θ ] n cos θ + n 1 n 2 sin 2 θ [ 1 + σ ( Ω ) cos θ ] ,
k 0 Δ GH = w p ϕ p θ + w s ϕ s θ ,
k 0 Δ IF = 4 a p a s a p 2 + a s 2 cot θ sin ( η ϕ p ϕ s 2 ) cos ( ϕ p ϕ s 2 ) ,

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