Lateral Displacement of a Light Beam at a Dielectric Interface*

Horowitz, B. R.; Tamir, T.

doi:10.1364/josa.61.000586

Cited by 257 publications

(141 citation statements)

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“…In 1970 [20], Horowitz and Tamir, by using a Fresnel approximation to analytically solve the integral determining the propagation of the transmitted beam, found, for the TE and TM lateral displacement, a closed expression in terms of parabolic-cylinder (Weber) functions. In the critical region, the Horowitz-Tamir formula, translated in our notation…”

Section: Discussionmentioning

confidence: 99%

“…(20). The formulations based on the Weber and modified Bessel functions give an analytical expression for the lateral displacement of the beam intensity maximum.…”

Section: Discussionmentioning

confidence: 99%

“…also thanks Silvânia A. Carvalho for stimulating conversations which motivated the study of an analytical expression for the GH shift in the critical region. Finally, the authors are greatly indebted to an anonymous referee for drawing their attention to the analytical work of Horowitz and Tamir [20], the paper comparing analytical formulas with experimental data [21], and the articles showing the effect of partial coherence in the GH shift [24,25].…”

Section: Acknowledgementsmentioning

confidence: 99%

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Closed-form expression for the Goos-Hänchen lateral displacement

Araújo

Maia

2016

Phys. Rev. A

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Abstract. The Artmann formula provides an accurate determination of the Goos-Hänchen lateral displacement in terms of the light wavelength, refractive index and incidence angle. In the total reflection region, this formula is widely used in the literature and confirmed by experiments. Nevertheless, for incidence at critical angle, it tends to infinity and numerical calculations are needed to reproduce the experimental data. In this paper, we overcome the divergence problem at critical angle and find, for Gaussian beams, a closed formula in terms of modified Bessel functions of the first kind. The formula is in excellent agreement with numerical calculations and reproduces, for incidence angles greater than critical ones, the Artmann formula. The closed form also allows one to understand how the breaking of symmetry in the angular distribution is responsible for the difference between measurements done by considering the maximum and the mean value of the beam intensity. The results obtained in this study clearly show the Goos-Hänchen lateral displacement dependence on the angular distribution shape of the incoming beam. Finally, we also present a brief comparison with experimental data and other analytical formulas found in the literature.

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Section: Discussionmentioning

confidence: 99%

“…(20). The formulations based on the Weber and modified Bessel functions give an analytical expression for the lateral displacement of the beam intensity maximum.…”

Section: Discussionmentioning

confidence: 99%

Section: Acknowledgementsmentioning

confidence: 99%

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Closed-form expression for the Goos-Hänchen lateral displacement

Araújo

Maia

2016

Phys. Rev. A

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“…When PCFs are reflected at the interface (z 1 = z 2 = z) between two media, each mode ψ m (for both TE and TM polarization) experiences a GH shift. Therefore the reflected CSD for a single mode ψ m , at the interface, can be formally written aswhere δθ m and ∆ m are the angular spread and the practical GH shift of the mth mode, respectively, and r(θ 0 , δθ m ) is the averaged reflection coefficient within δθ m around the incident angle θ 0 for the mth mode.Since δθ m may be very broad for a large m, the first-order Taylor expansion (FOTE) on the reflection coefficient r around θ 0 can fail [18]. Thus ∆ m are very different for…”

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confidence: 99%

Goos-Hänchen Shifts of Partially Coherent Light Fields

2013

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We investigate the Goos-Hänchen (GH) shifts of partially coherent fields (PCFs) by using the theory of coherence. We derive a formal expression for the GH shifts of PCFs in terms of Mercer's expansion, and then clearly demonstrate the dependence of the GH shift of each mode of PCFs on spatial coherence and beam width. We discuss the effect of spatial coherence on the resultant GH shifts, especially for the cases near the critical angles, such as totally reflection angle.PACS numbers: 42.50. Ar, 42.25.Gy, 42.30.Jf Goos-Hänchen (GH) Shift refers to a tiny (lateral) displacement, from the path expected from geometrical optics, upon total reflection [1]. This effect has been extended into other fields that involve the coherent-wave phenomena, such as neutron waves [2,3], electron waves [4,5], and spin waves [6]. It was explained by Artmann [7] that the different transverse wave vectors of a light beam undergo different phase changes and sum of these waves form a reflected beam with a lateral shift. Recently, it was shown [8] that the GH shift is the sum of Renard's conventional energy flux plus a self-interference shift. The self-interference shift originates from the interference between the incident and the reflected beams. Furthermore, it was discovered that the classical Fresnel formulas for laws of refraction and reflection of light are not applicable to partially coherent light [9]. These explanations indicate that the interference or coherence of light is very important to the GH shift.In 2008, we numerically showed the effect of spatial coherence on the change of the GH shift near the critical angles [10]. Later, an experiment [11] showed the large difference between the measured GH shift of a partially coherent LED light and the theoretical result of a coherent light. However, the very recent investigations [12][13][14][15][16] have raised an important issue "whether the spatial coherence of the partially coherent fields (PCFs) influences the GH shifts?" Although the exact numerical results, calculated from our previous theory [10], are in good agreement with the experimental data [13,14,16], it is necessary to reconsider this issue thoroughly and bring to light the role of spatial coherence on the GH shift.In this Letter, we use the exact theory of coherence to investigate the GH shift of PCFs. First we derive a formal expression to calculate the GH shift of PCFs in terms of the mode expansion of PCFs. Based on this expression, we explain the physical mechanism about the dependence of the GH shift on the spatial coherence and the beam width. Finally, we suggest a proposal for showing the new effect of the spatial coherence on the practical GH shift below the critical angles.First we derive the GH shift of PCFs based on the coherence theory [17]. For the two-dimensional PCFs, one usually uses the cross-spectral density (CSD), W (x 1 , z 1 ; x 2 , z 2 , ν), to describe its propagation, where (x 1 , z 1 ) and (x 2 , z 2 ) are the coordinates of the two points in the fields, and ν is the frequency of light....

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“…The Goos-Hänchen effect [1,2], which usually refers to the lateral shift of a totally reflected beam displaced from the path of geometrical optics, has been widely analyzed both theoretically [3][4][5] and experimentally [6][7][8][9]. The concept of Goos-Hänchen lateral shift has been expanded to partial reflection at arbitrary incident angle [10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Lateral Displacement of an Electromagnetic Beam Reflected From a Grounded Indefinite Uniaxial Slab

Kong¹,

Wu²,

Huang³

et al. 2008

PIER

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Abstract-A theoretical analysis of the lateral shift for an electromagnetic beam reflected from an uniaxial anisotropic slab coated with perfect conductor is presented. The analytic expression for the lateral shift is derived by using the stationary-phase approach, and the conditions for negative and positive lateral shifts are discussed. It is shown that the lateral shift depends not only on the slab thickness and the incident angle, but also on the constitutive parameters of the uniaxial medium. Enhancement and suppression of lateral shift are observed and are attributed to the interference between the reflected

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Lateral Displacement of a Light Beam at a Dielectric Interface*

Cited by 257 publications

References 5 publications

Closed-form expression for the Goos-Hänchen lateral displacement

Closed-form expression for the Goos-Hänchen lateral displacement

Goos-Hänchen Shifts of Partially Coherent Light Fields

Lateral Displacement of an Electromagnetic Beam Reflected From a Grounded Indefinite Uniaxial Slab

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