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Behavior of obliquely incident vector Bessel beams at planar interfaces Mohamed A. Salem* and Hakan Bağcı Division of Computer, Electrical and Mathematical Sciences and Engineering, 4700 King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia *Corresponding author: [email protected] Received February 19, 2013; revised April 16, 2013; accepted April 20, 2013; posted April 22, 2013 (Doc. ID 185528); published May 16, 2013 We investigate the behavior of full-vector electromagnetic Bessel beams obliquely incident at an interface between two electrically different media. We employ a Fourier transform domain representation of Bessel beams to determine their behavior upon reflection and transmission. This transform, which is geometric in nature, consists of elliptical support curves with complex weighting associated with them. The behavior of the scattered field at an interface is highly complex, owing to its full-vector nature; nevertheless, this behavior has a straightforward representation in the transform domain geometry. The analysis shows that the reflected field forms a different vector Bessel beam, but in general, the transmitted field cannot be represented as a Bessel beam. Nevertheless, using this approach, we demonstrate a method to propagate a Bessel beam in the refractive medium by launching a nonBessel beam at the interface. Several interesting phenomena related to the behavior of Bessel beams are illustrated, such as polarized reflection at Brewster’s angle incidence, and the Goos–Hänchen and Imbert–Federov shifts in the case of total reflection. © 2013 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (070.3185) Invariant optical fields; (080.0080) Geometric optics. http://dx.doi.org/10.1364/JOSAA.30.001172
1. INTRODUCTION Bessel beams are rapidly becoming essential elements in various applications, such as imaging, secure communications, and remote sensing. Recently, they have been successfully employed in 3D imaging of living cells [1], imaging in thick materials [2], nanostructure fabrication [3], virtual ghost imaging [4], and optical injection of living cells [5]. Bessel beams possess self-imaging properties [6], carry orbital angular momentum [7], and could transfer localized electromagnetic energy [8]. It is thus important to understand their behavior upon interaction with different structures and materials. One basic and canonical form of interaction concerns their behavior at a planar interface between two halfspaces with different electrical properties. Previous work has been conducted to characterize the behavior of Bessel beams as they pass through a dielectric-conducting medium interface [9] and through dielectric slabs at oblique angles of incidence [10]; however, this work focused only on the scalar zero-order Bessel beam. Goos–Hänchen (GH) and Imbert–Federov (IF) shifts in the case of total reflection of obliquely incident scalar Bessel beams under the paraxial propagation condition were discussed in [11]. For vector Bessel beams, IF shift and the change in their transverse structure under total reflection upon normal incidence were investigated in [12]. In this work, we start with a formal derivation of full-vector Bessel beams, which make use of the scalar Bessel beam solution introduced in [13]. Next, we provide an algebraic Fourier-transform-based representation of the obliquely incident Bessel beam and deduce simple relations congruent to Snell’s laws for reflection and refraction. We follow up on this pictorial representation with a rigorous derivation of 1084-7529/13/061172-08$15.00/0
the reflection and refraction coefficients for full-vector Bessel beams in the Fourier space. Two distinct cases are discussed in greater detail after that. The first case addresses the reflection and transmission of the incident field when the refracted field is propagating. In this case we additionally examine the behavior of the scattered field and its polarization upon incidence at Brewster’s angle, and we also examine the conditions under which the refracted field forms a Bessel beam. The second case addresses the behavior of the scattered field under the total reflection condition and the accompanying GH and IF shifts. We examine these shifts for oblique incidence on right-handed material and left-handed metamaterial half-spaces. Finally, we give the conclusions of this work and briefly discuss possible future extensions.
2. FULL-VECTOR BESSEL BEAMS Bessel beams are the fundamental solutions of the scalar homogeneous wave equation in the cylindrical coordinate system ρ; ϕ; z. The general representation of a Bessel beam of order n propagating in the z direction is given by ψ n ρ; ϕ; z; t AJ n kρ ρeinϕ eikz z−ωt :
(1)
Here J n x is ordinary Bessel function of first kind and order n, A is an arbitrary complex amplitude, ω is the angular frequency, and kρ and kz are the transversal and longitudinal components of the wavevector k. The wavevector components are more conveniently described in terms of the magnitude of the wavevector k and the “cone angle” ζ, such that kρ k sinζ and kz k cosζ. © 2013 Optical Society of America
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Next, we employ the scalar solution (1) to deduce the electromagnetic full-vector form. Yet, since assigning the scalar solution directly to a single component of the electric or magnetic field does not provide a proper solution to Maxwell’s equations [14], we adopt the approach presented in [15] to derive a transverse electric or transverse magnetic vector representation from a single vector potential. However, here we extend this approach to two vector potentials Πe r and Πh r to construct full-vector electromagnetic Bessel beams. The Lorenz gage condition relating a vector potential Πr to the scalar potential Φr under harmonic time dependence exp−iωt is given by Φr −ic∕kΠr with c the speed of light in the medium. The electric and magnetic fields are thus derived from the potentials as h c i Er −μ∇ × −i Πh r k ∇∇ · Πe r k2 Πe r; h c i Hr ϵ∇ × −i Πe r k ∇∇ · Πh r k2 Πh r;
csc2 ζ nμ ∂ − Ae i cosζAh J k sinζρ; k ρ ∂ρ n csc2 ζ n ∂ Eϕ − cosζAh − iμAe J k sinζρ; k ρ ∂ρ n Eρ
Ez Ae J n k sinζρ; csc2 ζ nϵ ∂ Ah i cosζAe J k sinζρ; Hρ k ρ ∂ρ n csc2 ζ n ∂ Hϕ − cosζAe iϵAh J k sinζρ; k ρ ∂ρ n H z Ah J n k sinζρ; where the factor expik cosζz nϕ − ωt is omitted throughout.
3. ANALYSIS OF OBLIQUE INCIDENCE
(2)
where ϵ and μ are the permittivity and permeability of the medium. The vector potentials can have nonzero divergence, thus permitting single component solutions satisfying the Helmholtz equation. Here, the nonzero z components of the vector potentials are both chosen to be scalar Bessel beam solutions of the same order but having different amplitudes Ae and Ah for Πe r and Πh r, respectively; i.e.,
Ae J n k sinζρeik cosζznϕ zˆ ; k2 sin2 ζ A Πh r 2 h2 J n k sinζρeik cosζznϕ zˆ ; k sin ζ Πe r
(3)
with zˆ the unit vector in the z direction. Inserting Eq. (3) into Eq. (2) yields the representation of the full-vector Bessel beam of order n, having the field components
x
x
A. Geometric Representation of the Spectra We start with a description of the geometric representation of the algebraic Fourier transform of oblique Bessel beams. For ease of analysis, the Cartesian coordinate system x; y; z is adopted with the corresponding transform domain kx ; ky ; kz . The following relations hold for oblique incidence at angle θ measured counterclockwise from the z axis: q x02 y02 ;
x0 x cosθ z sinθ;
ρ0
y0 y;
ϕ0 arctany0 ∕x0 ;
(4)
where the primed variables are the corresponding transformed coordinate variables from the oblique beam coordinate space into the reference coordinate system. Figure 1 depicts the spatial spectrum of a transverse section of oblique Bessel beams at z 0 with different angles of incidence. The transform reveals that the spectra have supports that lie on elliptic curves governed by the parametric equation kx σ ksinζ cosθ cosσ cosζ sinθ; ky σ k sinζ sinσ;
(5)
Normal incidence
z
z
x
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Oblique incidence Grazing incidence
ky
z
θ
( k , µk )
x'
kρ
kx
z'
(
χ
, µχ ) kz
Fig. 1. Illustration of an obliquely incident Bessel beam at different angles of incidence on a planar interface between two dielectric half-spaces (left), and the corresponding spectral support representation in the spatial Fourier domain (right), as given by Eq. (5).
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with σ varying between π. The eccentricity of the ellipse is given by ε sinθ, and its center is shifted by k cosζ sinθ along the kx axis. This shift along the kx axis is due to the phase component expik cosζz0 along the x–y plane. Here we note two important points. First, the order of the Bessel beam is expressed in the complex weight associated with the spectral support curve but does not affect the shape of the support curve itself; therefore the spectral support described above is valid for all obliquely incident Bessel beams of any order n. Second, for the full-vector Bessel beam, all field components have the same spectral support curve, since the argument of their Bessel functions and their z0 -directed harmonic components is identical. To describe the propagation of the Bessel beam, it is necessary to consider the third component of the wavevector, kz . In the case of the incident beam, the transformed spectral support (5) is projected on a hemisphere centered at the origin of the kx –ky plane and has a radius equal to k. In terms of the parameter σ, it is described as kz σ
q k2 − k2x σ k2y σ;
(6)
where the positive value of the root is taken. The projection of the support curve on the kx − kz plane, as shown in Fig. 2, reveals that it lies on a plane inclined with the angle θ. This pictorial illustration is especially beneficial when considering the behavior of the incident beam at a planar interface. Using the spectral representation, and knowing that the transverse wavevector components are conserved upon incidence, it directly follows that only the corresponding projection hemispheres will change in the cases of reflection and refraction. For the reflected field, the only difference is that kz acquires negative values [choosing the negative value of the root in Eq. (6)]. This translates into a hemisphere with radius k, and its normal is in the negative direction. The shape of
the spectral support curve does not change, as shown in Fig. 2, and hence the reflected field forms an undistorted Bessel beam propagating away from the interface with angle −θ, which is in agreement with Snell’s reflection law. In the refraction medium with a wavevector χ, the radius of the projection hemisphere changes to χ. This corresponding projection of the support curve does not lie in one plane; i.e., the phase front of the refracted field is not planar, so it does not form an undistorted Bessel beam. Nevertheless, if the field behavior at the spectral points with major phase coherence is considered (at maximum jky j), the angle at which most of the refracted field energy propagates, ϑ, can be determined as nk sinϑ q sinθ; 2 2 nχ nχ − n2k tan2 ζ
where nk and nχ are the refractive indices in the medium of incidence and the medium of refraction, respectively. It is noteworthy that ϑ is dependent on ζ, and in the special case where ζ 0 (i.e., the Bessel beam reduces to a plane wave), the relation (7) reduces to Snell’s refraction law. A depiction of the distortion in the spectral support for the field refracted into a medium denser than the medium of incidence is shown in Fig. 2. Moreover, this geometric Fourier representation confirms that normally incident Bessel beams will undergo total reflection if k sinζ > χ, as previously reported in [12,16]. B. Formulation of the Reflection and Transmission Coefficients The rigorous description of the scattered field is obtained by matching the tangential field components at the planar interface. For this, the incident field components at z 0 are converted from cylindrical coordinates into Cartesian coordinates using Eq. (4), and the field components are expressed in terms of Cartesian components as
E E ρ ϕ x; y ρ0 ; ϕ0 cosϕ0 − ρ0 ; ϕ0 sinϕ0 Hρ Hϕ Hx Ez ρ0 ; ϕ0 sinθ; × cosθ − Hz E E E y ρ ϕ x; y ρ0 ; ϕ0 sinϕ0 ρ0 ; ϕ0 cosϕ0 : Hρ Hϕ Hy
1.5 Incident Reflected Refracted
1
(7)
kz/k
0.5
Ex
0
The tangential field components follow naturally and are more easily found in the transform space as -0.5
-1
-1
-0.5
0
0.5
1
E~ ix k E~ rx k E~ tx k;
~ ix k H ~ rx k H ~ tx k; H
E~ iy k E~ ry k E~ ty k;
~ iy k H ~ ry k H ~ ty k; H
kx/k Fig. 2. Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kx –kz plane as given by Eq. (6). The incident Bessel beam has a cone anglepζ π∕6 and is incident at angle θ π∕4 onto a half-space with χ 2k. The figure shows the deformation in the support curve of the refracted field resulting in a diverging refracted beam.
with the superscripts i, r, and t designating the incident, reflected, and refracted fields, respectively. The reflected and refracted fields in the transform space are thus fully described ~ e∕h and in terms of the reflection and refraction coefficients G T~ e∕h as
M. A. Salem and H. Bağcı
Vol. 30, No. 6 / June 2013 / J. Opt. Soc. Am. A
~ e kx kzk − G ~ h ky μk ωeikx xky y−kzk z ; E~ rx G
beams incident at this angle. The second case involves an incident “distorted” field that results in a refracted Bessel beam. This case is of interest from an applied point view, since usually in practical applications the properties of the medium where the Bessel beam is launched are different from those of the medium where the interaction with a Bessel beam is sought after. Hence, it is useful to be able to launch an undistorted Bessel beam into the refractive medium.
~ h kx kzk G ~ e ky ϵk ωeikx xky y−kzk z ; ~ rx G H ~ e ky kzk G ~ h kx μk ωeikx xky y−kzk z ; E~ ry G ~ h ky kzk − G ~ e kx ϵk ωeikx xky y−kzk z ; ~ ry G H E~ tx −T~ e kx kzχ T~ h ky μχ ωeikx xky ykzχ z ; ~ tx −T~ h kx kzχ T~ e ky ϵχ ωeikx xky ykzχ z ; H E~ ty −T~ e ky kzχ T~ h kx μχ ωeikx xky ykzχ z ; ~ ty −T~ h ky kzχ T~ e kx ϵχ ωeikx xky ykzχ z ; H
(8)
q q with kzk k2 − k2x k2y , kzχ χ 2 − k2x k2y , and the positive values of the square roots are to be taken. The coeffi~ e∕h and T~ e∕h are given in terms of the incident cients G beam spectrum as ~ e − kzχ H i− ϵχ E i ∕De ; G ω ~ h − − kzχ E i− μχ H i ∕Dh ; G ω k T~ e − − zk H i− ϵk E i ∕De ; ω k T~ h − zk Ei− μk H i ∕Dh ; ω
(9)
~ x− where E − ky E~ x − kx E~ y , E ky E~ x ky E~ y , H − ky H 2 2 ~ y , H kx H ~ x ky H ~ y , De kx ky ϵk kzχ ϵχ kzk , kx H
Dh k2x k2y μk kzχ μχ kzk , and the subscripts k and χ refer to the incidence and refraction media, respectively. Finally, the z-directed field components are determined by Maxwell’s equations as ~ e k2x k2y eikx xky y−kzk z ; E~ rz G ~ h k2x k2y eikx xky y−kzk z ; ~ rz G H E~ tz T~ e k2x k2y eikx xky ykzχ z ; ~ tz T~ h k2x k2y eikx xky ykzχ z : H
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(10)
The field expressions in space are found by taking the inverse Fourier transform of Eqs. (8) and (10). This completes the analysis of the oblique incidence of full-vector Bessel beams at half-spaces.
4. REFLECTION AND TRANSMISSION CASE Here, we consider the case where χ > k; thus the field in the refraction medium is propagating. In particular, we will consider two specific cases; the first is oblique incidence at Brewster’s angle. This case is of interest since in the case of unpolarized plane wave incidence, the reflected field is perfectly polarized in a certain direction, or equivalently, fields with certain polarization are completely refracted without reflection. Applications that take advantage of this phenomenon are numerous; however, Brewster windows are especially important in gas laser devices [17]. Thus in order to advance Bessel beam lasers, it is essential to understand the reflection and refraction of Bessel
A. Incidence at Brewster’s Angle Brewster’s angle θB is defined by the relation θB ϑ π∕2. Using Eq. (7), Brewster’s angle for Bessel beams is thus given by
tanθB
q n2χ n2χ − n2k tan2 ζ nk
:
(11)
We should note here that this definition is strictly the polarization angle for the spectral points along the major axis of the support elliptical curve, which usually are the spectral points with maximum energy contribution. However, depending on the excitation amplitudes of the beam, the maximum spectral energy concentration may not be at these points and accordingly the maximum polarization angle (for electric or magnetic fields) may vary. Figure 3 depicts the reflection and refraction of E x and E y of a full-vector Bessel beam incident at polarization angle θB π∕2.7433 calculated using Eq. (11). The figure depicts the condition with maximum energy transmission into the refractive medium for the x-polarized electric field component. A comparison between E x and E y at the interface reveals that the reflected electric field is mostly y polarized. B. Refracted Bessel Beam Utilizing the same geometric approach described in Section 3.A, we devise a scheme for producing a Bessel beam in the refraction medium using a non-Bessel incident field. First, we note that the spectral support of the Bessel beam is deformed upon refraction due to the difference in the wave number between the incidence and refraction media. Next, we use this information to specify the spectrum of the incident field such that it is converted into a Bessel beam spectrum upon refraction. This procedure is carried out over two steps. In the first step, the required Bessel beam spectrum is specified in the refraction medium, and in the second step, this support curve is projected on the kz hemisphere of the incidence medium. The schematic in Fig. 4 shows the deformed incident spectral support curve resulting in a Bessel beam support curve in the refraction medium. This approach is always successful when χ < k; however, extra care should to be taken when χ > k, since the required spectrum may not be fully mapped onto the kz hemisphere of the incidence medium. To further specify the properties of the refracted Bessel beam, the procedure described above must be accompanied with the relevant computation of the necessary excitation amplitudes of the incident field. This is carried out in a straightforward fashion using Eqs. (8) and (9). Solving for the incident field, we get
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Fig. 3. Bessel beam incident at Brewster’s angle θ π∕2.7433 onto a dielectric half-space with ϵχ 3.9ϵk , μχ μk , and Ae −iAh 1. The reflection and refraction of the E x component in the x–z plane are shown in (a), and the cross section of its reflected field is shown in (b). The reflection and refraction of the Ey component in the x–z plane are shown in (c), and the cross section of its reflected field is shown in (d). The figure clearly shows that the reflected field is mostly polarized in the y direction. Note that the field values are normalized with respect to their respective incident fields.
E~ ix −I e kx kzk − I h ky μk ωeikx xky ykzk z ;
with the excitation coefficients
~ ix −I h kx kzk I e ky ϵk ωeikx xky ykzk z ; H E~ iy −I e ky kzk I h kx μk ωeikx xky ykzk z ; ~ iy −I h ky kzk − I e kx ϵk ωeikx xky ykzk z ; H
(12)
k ~ e ; I h − kzk Et− μk H t ∕D ~ h; I e − − zk H t− ϵk E t ∕D ω ω
1.5 Incident Reflected Refracted
1
kz/k
0.5 0 -0.5 -1
-1
-0.5
0
0.5
1
kx/k
(a)
(b)
Fig. 4. (a) Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kx –kz plane as given by Eq. (6). The support curve of the incident field is deformed such that the refracted field forms a Bessel beam with cone angle p ζ π∕6 and refraction angle ϑ π∕4 through a half-space with χ 2k. Note that for this cone angle and material, ϑ π∕4 is the maximum possible refraction angle for a complete Bessel beam, since greater angles would drive the incident spectrum into the evanescence region. (b) Bessel beam refracted at angle ϑ π∕6 into a medium with ϵχ 3.9ϵk and μχ μk . The incident and reflected fields do not form a Bessel beam; however, the incident field is converging at the interface.
M. A. Salem and H. Bağcı
~ e 2k2x k2y kzk ϵk and D ~ h 2k2x k2y kzk μk , and the where D refracted field spectra are those of the required Bessel beam. Figure 4 depicts the refraction of a full-vector Bessel beam at angle ϑ π∕6 due to an incident field determined by Eq. (12).
5. TOTAL REFLECTION AND SHIFT CASE Here, we consider the case where the refracted field is evanescent, or predominantly evanescent. Several conditions exist in which these criteria are met, resulting in total reflection for the whole or the greater part of the spectrum. In general, the necessary condition that must hold in order to have no beam-like propagating field in the refraction p medium is χ < k sin2 ζ cos2 ζ sin2 θ, where the media properties, the angle of incidence, θ, and the cone angle, ζ, are involved. In [12], the GH and IF shifts occurring upon normal incidence of a vector Bessel beam were qualitatively discussed in terms of the change in the transverse Poynting vector components. It was argued that the azimuthal component of the Poynting vector S ϕ transfers the energy along the beam ring and is responsible for the GH shift. The GH shift in this case manifests itself in the form of a beam rotation that cannot be visually seen. The radial component of the Poynting vector S ρ transfers the energy in the radial direction at the interface and is responsible for the IF shift. The IF shift in this case manifests itself in the form of a change in the beam spot size. The longitudinal component of the Poynting vector S z is equal to zero, since the evanescent Bessel beam does not carry energy along its axis.
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In the case of oblique incidence, two types of GH shifts take place; the first is a lateral shift along the direction of incidence (x axis), and the second is a rotational shift. To identify the lateral shift, we recall that for total reflection, the reflection coefficients of the field are complex with a magnitude equal to unity. Accordingly, if the reflection coefficients could be ~ expiγ, then the lateral GH is given written in the form G by Δx −dγ∕dkx . From Eq. (9), it is obvious that the two quantities De and Dh are the quantities responsible for the shifts. It also becomes evident that there are two distinct lateral shifts in the case of full-vector Bessel beam total reflection, which are given by Δxe
kx ϵk ϵχ k2 − χ 2 ; kzk jkzχ jϵ2χ k2zk − ϵk k2zχ
Δxh
kx μk μχ k2 − χ 2 : kzk jkzχ jμ2χ k2zk − μ2k k2zχ (13)
The excitation amplitudes and the refraction medium properties are thus the determining factors, of which shift is the dominant one. Additionally, for the field reflected from a right-handed material half-space, the phase shift is negative, resulting in a positive displacement along the x axis; whereas in the case of left-handed metamaterial half-space, a positive phase shift occurs and accordingly the field exhibits a negative displacement along the x axis. The refracted evanescent fields exhibit the same lateral shifts, except in the opposite direction. Quantification of the rotational GH shift and the IF shift is more involved, since both are related to phase change with respect to ky . We also note that in the case of oblique
Fig. 5. Comparison between the scattered fields upon incidence under total reflection condition. In (a), (b), and (c), the refraction medium is a right-handed material with ϵχ ϵk ∕3.9 and μχ μk , and in (d), (e), and (f), the refraction medium is a left-handed metamaterial with ϵχ −ϵk ∕3.9 and μχ −μk . In (a), a positive lateral GH shift along the x axis in the reflected Ex is visible in the x–z plane, in contrast to a negative GH lateral shift for the same component in (d). The lateral and rotational GH shifts for Ex are visible in (b) and (c), while these shifts are mirrored in (e) and (f). The change in the transverse field structure due to the IF shift is visible in (b), (c), (e), and (f).
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incidence, an “unspoiled” IF shift will not occur and the change in the spot size will suffer from unevenness. The reason for this unevenness is that the spectral support is symmetric with respect to the kx axis, but asymmetric with respect to the ky axis. The asymmetry across the ky axis results in a net displacement along the x axis (lateral GH shift), since the whole spectral support must be in the positive kx region to exhibit total reflection. On the other hand, the additional phase shift introduced by the reflection and refraction coefficients coupled by the symmetry of the spectrum support across the kx axis results in rotation (rotational GH shift) and change in the shape and size of the beam cross section (IF shift). Similarly, the phase change with respect to ky due to the quantities De and Dh is defined as Δy −dγ∕dky and is identical to Eq. (13) after exchanging kx ↔ky . Figure 5 depicts the shifts for oblique incidence upon two different half-spaces; the first is composed of a right-handed material, and the second is composed of a left-handed metamaterial. The figure clearly shows that the GH shifts in the case of the metamaterial half-space are reversed compared to the other case. The IF shift did not change, though, since it does not depend on the sign of the phase shift (direction of the displacement), but instead depends on its magnitude, which is the same for both cases.
6. CONCLUSION We have presented a pictorial illustrative method that describes the behavior of obliquely incident Bessel beams at a planar interface between two dielectric half-spaces with different material properties. This method is based on the algebraic spatial Fourier transform of the Bessel beam, where the spectral support of the Bessel beam is given by elliptic curves with complex weightings. Using this method, we deduced simple relations for the reflection and refraction of the field congruent to Snell’s laws. We also rigorously derived expressions for the scattered field after matching its tangential components at the interface. Final expressions of the scattered field in the spatial domain are determined by taking the inverse Fourier transform of the derived spectral field expressions. Using the obtained expressions of the scattered field, we deduced an expression for the polarization angle (Brewster’s angle) for obliquely incident Bessel beams and investigated the behavior of the scattered field upon incidence at such angle. We showed that the reflected field cannot be perfectly polarized due to the nature of the Bessel beam spectrum. Furthermore, we constructed a scheme to produce propagating Bessel beams in the refraction medium. This scheme requires that the incident field has a deformed spectral support curve and thus cannot be propagation-invariantlike Bessel beams. Finally, we have investigated the GH and IF shifts that occur when incident Bessel beams undergo total or almost total reflection. We have also shown that the GH shifts in the case of incidence upon a left-handed metamaterial half-space are the mirror images of the corresponding shifts if the half-space is composed of a right-handed material. The derived formulations and obtained results are principally important in fundamental understanding of the interaction of Bessel beams at planar interfaces, and applications that involve their scattering. The presented
M. A. Salem and H. Bağcı
simplified formulations of the refraction angle, the polarization angle, and the qualitative description of the scattered field behavior based on its spectral support provide practical and quick tools for understanding and engineering the interaction of Bessel beams. On the other hand, the rigorous formulation of the scattered field in the Fourier space provides a computationally inexpensive method to accurately determine the exact field in space. The work presented here lends itself to further extensions. Such extensions include investigating the behavior of obliquely incident vector Bessel beams on dielectric slabs and multilayer structures. This is particularly of interest in practical generation of laser Bessel beams, since gas laser devices rely on Brewster windows, which essentially are dielectric slabs with finite extension. Also the interaction of Bessel beams with biological tissues could be modeled as interaction with multilayer structures. This work could also be extended to pulsed Bessel beams and X waves, which are a chromatic collection of Bessel beams with extra conditions governing the relation between their spatiotemporal spectra.
ACKNOWLEDGMENTS This work was supported in part by the Center for Uncertainty Quantification in Computational Science and Engineering at King Abdullah University of Science and Engineering (KAUST).
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M. A. Salem and H. Bağcı 12. A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–-Fedorov shift and intensity transformation,” J. Opt. Pure Appl. Opt. 10, 075006 (2008). 13. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). 14. S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
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15. M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011). 16. M. A. Salem and H. Bağci, “Reflection and transmission of normally incident full-vector X waves on planar interfaces,” J. Opt. Soc. Am. A 29, 139–152 (2012). 17. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 589–601.
J. Opt. Soc. Am. A / Vol. 30, No. 6 / June 2013
M. A. Salem and H. Bağcı
Behavior of obliquely incident vector Bessel beams at planar interfaces Mohamed A. Salem* and Hakan Bağcı Division of Computer, Electrical and Mathematical Sciences and Engineering, 4700 King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia *Corresponding author: [email protected] Received February 19, 2013; revised April 16, 2013; accepted April 20, 2013; posted April 22, 2013 (Doc. ID 185528); published May 16, 2013 We investigate the behavior of full-vector electromagnetic Bessel beams obliquely incident at an interface between two electrically different media. We employ a Fourier transform domain representation of Bessel beams to determine their behavior upon reflection and transmission. This transform, which is geometric in nature, consists of elliptical support curves with complex weighting associated with them. The behavior of the scattered field at an interface is highly complex, owing to its full-vector nature; nevertheless, this behavior has a straightforward representation in the transform domain geometry. The analysis shows that the reflected field forms a different vector Bessel beam, but in general, the transmitted field cannot be represented as a Bessel beam. Nevertheless, using this approach, we demonstrate a method to propagate a Bessel beam in the refractive medium by launching a nonBessel beam at the interface. Several interesting phenomena related to the behavior of Bessel beams are illustrated, such as polarized reflection at Brewster’s angle incidence, and the Goos–Hänchen and Imbert–Federov shifts in the case of total reflection. © 2013 Optical Society of America OCIS codes: (240.0240) Optics at surfaces; (070.3185) Invariant optical fields; (080.0080) Geometric optics. http://dx.doi.org/10.1364/JOSAA.30.001172
1. INTRODUCTION Bessel beams are rapidly becoming essential elements in various applications, such as imaging, secure communications, and remote sensing. Recently, they have been successfully employed in 3D imaging of living cells [1], imaging in thick materials [2], nanostructure fabrication [3], virtual ghost imaging [4], and optical injection of living cells [5]. Bessel beams possess self-imaging properties [6], carry orbital angular momentum [7], and could transfer localized electromagnetic energy [8]. It is thus important to understand their behavior upon interaction with different structures and materials. One basic and canonical form of interaction concerns their behavior at a planar interface between two halfspaces with different electrical properties. Previous work has been conducted to characterize the behavior of Bessel beams as they pass through a dielectric-conducting medium interface [9] and through dielectric slabs at oblique angles of incidence [10]; however, this work focused only on the scalar zero-order Bessel beam. Goos–Hänchen (GH) and Imbert–Federov (IF) shifts in the case of total reflection of obliquely incident scalar Bessel beams under the paraxial propagation condition were discussed in [11]. For vector Bessel beams, IF shift and the change in their transverse structure under total reflection upon normal incidence were investigated in [12]. In this work, we start with a formal derivation of full-vector Bessel beams, which make use of the scalar Bessel beam solution introduced in [13]. Next, we provide an algebraic Fourier-transform-based representation of the obliquely incident Bessel beam and deduce simple relations congruent to Snell’s laws for reflection and refraction. We follow up on this pictorial representation with a rigorous derivation of 1084-7529/13/061172-08$15.00/0
the reflection and refraction coefficients for full-vector Bessel beams in the Fourier space. Two distinct cases are discussed in greater detail after that. The first case addresses the reflection and transmission of the incident field when the refracted field is propagating. In this case we additionally examine the behavior of the scattered field and its polarization upon incidence at Brewster’s angle, and we also examine the conditions under which the refracted field forms a Bessel beam. The second case addresses the behavior of the scattered field under the total reflection condition and the accompanying GH and IF shifts. We examine these shifts for oblique incidence on right-handed material and left-handed metamaterial half-spaces. Finally, we give the conclusions of this work and briefly discuss possible future extensions.
2. FULL-VECTOR BESSEL BEAMS Bessel beams are the fundamental solutions of the scalar homogeneous wave equation in the cylindrical coordinate system ρ; ϕ; z. The general representation of a Bessel beam of order n propagating in the z direction is given by ψ n ρ; ϕ; z; t AJ n kρ ρeinϕ eikz z−ωt :
(1)
Here J n x is ordinary Bessel function of first kind and order n, A is an arbitrary complex amplitude, ω is the angular frequency, and kρ and kz are the transversal and longitudinal components of the wavevector k. The wavevector components are more conveniently described in terms of the magnitude of the wavevector k and the “cone angle” ζ, such that kρ k sinζ and kz k cosζ. © 2013 Optical Society of America
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Next, we employ the scalar solution (1) to deduce the electromagnetic full-vector form. Yet, since assigning the scalar solution directly to a single component of the electric or magnetic field does not provide a proper solution to Maxwell’s equations [14], we adopt the approach presented in [15] to derive a transverse electric or transverse magnetic vector representation from a single vector potential. However, here we extend this approach to two vector potentials Πe r and Πh r to construct full-vector electromagnetic Bessel beams. The Lorenz gage condition relating a vector potential Πr to the scalar potential Φr under harmonic time dependence exp−iωt is given by Φr −ic∕kΠr with c the speed of light in the medium. The electric and magnetic fields are thus derived from the potentials as h c i Er −μ∇ × −i Πh r k ∇∇ · Πe r k2 Πe r; h c i Hr ϵ∇ × −i Πe r k ∇∇ · Πh r k2 Πh r;
csc2 ζ nμ ∂ − Ae i cosζAh J k sinζρ; k ρ ∂ρ n csc2 ζ n ∂ Eϕ − cosζAh − iμAe J k sinζρ; k ρ ∂ρ n Eρ
Ez Ae J n k sinζρ; csc2 ζ nϵ ∂ Ah i cosζAe J k sinζρ; Hρ k ρ ∂ρ n csc2 ζ n ∂ Hϕ − cosζAe iϵAh J k sinζρ; k ρ ∂ρ n H z Ah J n k sinζρ; where the factor expik cosζz nϕ − ωt is omitted throughout.
3. ANALYSIS OF OBLIQUE INCIDENCE
(2)
where ϵ and μ are the permittivity and permeability of the medium. The vector potentials can have nonzero divergence, thus permitting single component solutions satisfying the Helmholtz equation. Here, the nonzero z components of the vector potentials are both chosen to be scalar Bessel beam solutions of the same order but having different amplitudes Ae and Ah for Πe r and Πh r, respectively; i.e.,
Ae J n k sinζρeik cosζznϕ zˆ ; k2 sin2 ζ A Πh r 2 h2 J n k sinζρeik cosζznϕ zˆ ; k sin ζ Πe r
(3)
with zˆ the unit vector in the z direction. Inserting Eq. (3) into Eq. (2) yields the representation of the full-vector Bessel beam of order n, having the field components
x
x
A. Geometric Representation of the Spectra We start with a description of the geometric representation of the algebraic Fourier transform of oblique Bessel beams. For ease of analysis, the Cartesian coordinate system x; y; z is adopted with the corresponding transform domain kx ; ky ; kz . The following relations hold for oblique incidence at angle θ measured counterclockwise from the z axis: q x02 y02 ;
x0 x cosθ z sinθ;
ρ0
y0 y;
ϕ0 arctany0 ∕x0 ;
(4)
where the primed variables are the corresponding transformed coordinate variables from the oblique beam coordinate space into the reference coordinate system. Figure 1 depicts the spatial spectrum of a transverse section of oblique Bessel beams at z 0 with different angles of incidence. The transform reveals that the spectra have supports that lie on elliptic curves governed by the parametric equation kx σ ksinζ cosθ cosσ cosζ sinθ; ky σ k sinζ sinσ;
(5)
Normal incidence
z
z
x
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Oblique incidence Grazing incidence
ky
z
θ
( k , µk )
x'
kρ
kx
z'
(
χ
, µχ ) kz
Fig. 1. Illustration of an obliquely incident Bessel beam at different angles of incidence on a planar interface between two dielectric half-spaces (left), and the corresponding spectral support representation in the spatial Fourier domain (right), as given by Eq. (5).
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with σ varying between π. The eccentricity of the ellipse is given by ε sinθ, and its center is shifted by k cosζ sinθ along the kx axis. This shift along the kx axis is due to the phase component expik cosζz0 along the x–y plane. Here we note two important points. First, the order of the Bessel beam is expressed in the complex weight associated with the spectral support curve but does not affect the shape of the support curve itself; therefore the spectral support described above is valid for all obliquely incident Bessel beams of any order n. Second, for the full-vector Bessel beam, all field components have the same spectral support curve, since the argument of their Bessel functions and their z0 -directed harmonic components is identical. To describe the propagation of the Bessel beam, it is necessary to consider the third component of the wavevector, kz . In the case of the incident beam, the transformed spectral support (5) is projected on a hemisphere centered at the origin of the kx –ky plane and has a radius equal to k. In terms of the parameter σ, it is described as kz σ
q k2 − k2x σ k2y σ;
(6)
where the positive value of the root is taken. The projection of the support curve on the kx − kz plane, as shown in Fig. 2, reveals that it lies on a plane inclined with the angle θ. This pictorial illustration is especially beneficial when considering the behavior of the incident beam at a planar interface. Using the spectral representation, and knowing that the transverse wavevector components are conserved upon incidence, it directly follows that only the corresponding projection hemispheres will change in the cases of reflection and refraction. For the reflected field, the only difference is that kz acquires negative values [choosing the negative value of the root in Eq. (6)]. This translates into a hemisphere with radius k, and its normal is in the negative direction. The shape of
the spectral support curve does not change, as shown in Fig. 2, and hence the reflected field forms an undistorted Bessel beam propagating away from the interface with angle −θ, which is in agreement with Snell’s reflection law. In the refraction medium with a wavevector χ, the radius of the projection hemisphere changes to χ. This corresponding projection of the support curve does not lie in one plane; i.e., the phase front of the refracted field is not planar, so it does not form an undistorted Bessel beam. Nevertheless, if the field behavior at the spectral points with major phase coherence is considered (at maximum jky j), the angle at which most of the refracted field energy propagates, ϑ, can be determined as nk sinϑ q sinθ; 2 2 nχ nχ − n2k tan2 ζ
where nk and nχ are the refractive indices in the medium of incidence and the medium of refraction, respectively. It is noteworthy that ϑ is dependent on ζ, and in the special case where ζ 0 (i.e., the Bessel beam reduces to a plane wave), the relation (7) reduces to Snell’s refraction law. A depiction of the distortion in the spectral support for the field refracted into a medium denser than the medium of incidence is shown in Fig. 2. Moreover, this geometric Fourier representation confirms that normally incident Bessel beams will undergo total reflection if k sinζ > χ, as previously reported in [12,16]. B. Formulation of the Reflection and Transmission Coefficients The rigorous description of the scattered field is obtained by matching the tangential field components at the planar interface. For this, the incident field components at z 0 are converted from cylindrical coordinates into Cartesian coordinates using Eq. (4), and the field components are expressed in terms of Cartesian components as
E E ρ ϕ x; y ρ0 ; ϕ0 cosϕ0 − ρ0 ; ϕ0 sinϕ0 Hρ Hϕ Hx Ez ρ0 ; ϕ0 sinθ; × cosθ − Hz E E E y ρ ϕ x; y ρ0 ; ϕ0 sinϕ0 ρ0 ; ϕ0 cosϕ0 : Hρ Hϕ Hy
1.5 Incident Reflected Refracted
1
(7)
kz/k
0.5
Ex
0
The tangential field components follow naturally and are more easily found in the transform space as -0.5
-1
-1
-0.5
0
0.5
1
E~ ix k E~ rx k E~ tx k;
~ ix k H ~ rx k H ~ tx k; H
E~ iy k E~ ry k E~ ty k;
~ iy k H ~ ry k H ~ ty k; H
kx/k Fig. 2. Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kx –kz plane as given by Eq. (6). The incident Bessel beam has a cone anglepζ π∕6 and is incident at angle θ π∕4 onto a half-space with χ 2k. The figure shows the deformation in the support curve of the refracted field resulting in a diverging refracted beam.
with the superscripts i, r, and t designating the incident, reflected, and refracted fields, respectively. The reflected and refracted fields in the transform space are thus fully described ~ e∕h and in terms of the reflection and refraction coefficients G T~ e∕h as
M. A. Salem and H. Bağcı
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~ e kx kzk − G ~ h ky μk ωeikx xky y−kzk z ; E~ rx G
beams incident at this angle. The second case involves an incident “distorted” field that results in a refracted Bessel beam. This case is of interest from an applied point view, since usually in practical applications the properties of the medium where the Bessel beam is launched are different from those of the medium where the interaction with a Bessel beam is sought after. Hence, it is useful to be able to launch an undistorted Bessel beam into the refractive medium.
~ h kx kzk G ~ e ky ϵk ωeikx xky y−kzk z ; ~ rx G H ~ e ky kzk G ~ h kx μk ωeikx xky y−kzk z ; E~ ry G ~ h ky kzk − G ~ e kx ϵk ωeikx xky y−kzk z ; ~ ry G H E~ tx −T~ e kx kzχ T~ h ky μχ ωeikx xky ykzχ z ; ~ tx −T~ h kx kzχ T~ e ky ϵχ ωeikx xky ykzχ z ; H E~ ty −T~ e ky kzχ T~ h kx μχ ωeikx xky ykzχ z ; ~ ty −T~ h ky kzχ T~ e kx ϵχ ωeikx xky ykzχ z ; H
(8)
q q with kzk k2 − k2x k2y , kzχ χ 2 − k2x k2y , and the positive values of the square roots are to be taken. The coeffi~ e∕h and T~ e∕h are given in terms of the incident cients G beam spectrum as ~ e − kzχ H i− ϵχ E i ∕De ; G ω ~ h − − kzχ E i− μχ H i ∕Dh ; G ω k T~ e − − zk H i− ϵk E i ∕De ; ω k T~ h − zk Ei− μk H i ∕Dh ; ω
(9)
~ x− where E − ky E~ x − kx E~ y , E ky E~ x ky E~ y , H − ky H 2 2 ~ y , H kx H ~ x ky H ~ y , De kx ky ϵk kzχ ϵχ kzk , kx H
Dh k2x k2y μk kzχ μχ kzk , and the subscripts k and χ refer to the incidence and refraction media, respectively. Finally, the z-directed field components are determined by Maxwell’s equations as ~ e k2x k2y eikx xky y−kzk z ; E~ rz G ~ h k2x k2y eikx xky y−kzk z ; ~ rz G H E~ tz T~ e k2x k2y eikx xky ykzχ z ; ~ tz T~ h k2x k2y eikx xky ykzχ z : H
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(10)
The field expressions in space are found by taking the inverse Fourier transform of Eqs. (8) and (10). This completes the analysis of the oblique incidence of full-vector Bessel beams at half-spaces.
4. REFLECTION AND TRANSMISSION CASE Here, we consider the case where χ > k; thus the field in the refraction medium is propagating. In particular, we will consider two specific cases; the first is oblique incidence at Brewster’s angle. This case is of interest since in the case of unpolarized plane wave incidence, the reflected field is perfectly polarized in a certain direction, or equivalently, fields with certain polarization are completely refracted without reflection. Applications that take advantage of this phenomenon are numerous; however, Brewster windows are especially important in gas laser devices [17]. Thus in order to advance Bessel beam lasers, it is essential to understand the reflection and refraction of Bessel
A. Incidence at Brewster’s Angle Brewster’s angle θB is defined by the relation θB ϑ π∕2. Using Eq. (7), Brewster’s angle for Bessel beams is thus given by
tanθB
q n2χ n2χ − n2k tan2 ζ nk
:
(11)
We should note here that this definition is strictly the polarization angle for the spectral points along the major axis of the support elliptical curve, which usually are the spectral points with maximum energy contribution. However, depending on the excitation amplitudes of the beam, the maximum spectral energy concentration may not be at these points and accordingly the maximum polarization angle (for electric or magnetic fields) may vary. Figure 3 depicts the reflection and refraction of E x and E y of a full-vector Bessel beam incident at polarization angle θB π∕2.7433 calculated using Eq. (11). The figure depicts the condition with maximum energy transmission into the refractive medium for the x-polarized electric field component. A comparison between E x and E y at the interface reveals that the reflected electric field is mostly y polarized. B. Refracted Bessel Beam Utilizing the same geometric approach described in Section 3.A, we devise a scheme for producing a Bessel beam in the refraction medium using a non-Bessel incident field. First, we note that the spectral support of the Bessel beam is deformed upon refraction due to the difference in the wave number between the incidence and refraction media. Next, we use this information to specify the spectrum of the incident field such that it is converted into a Bessel beam spectrum upon refraction. This procedure is carried out over two steps. In the first step, the required Bessel beam spectrum is specified in the refraction medium, and in the second step, this support curve is projected on the kz hemisphere of the incidence medium. The schematic in Fig. 4 shows the deformed incident spectral support curve resulting in a Bessel beam support curve in the refraction medium. This approach is always successful when χ < k; however, extra care should to be taken when χ > k, since the required spectrum may not be fully mapped onto the kz hemisphere of the incidence medium. To further specify the properties of the refracted Bessel beam, the procedure described above must be accompanied with the relevant computation of the necessary excitation amplitudes of the incident field. This is carried out in a straightforward fashion using Eqs. (8) and (9). Solving for the incident field, we get
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Fig. 3. Bessel beam incident at Brewster’s angle θ π∕2.7433 onto a dielectric half-space with ϵχ 3.9ϵk , μχ μk , and Ae −iAh 1. The reflection and refraction of the E x component in the x–z plane are shown in (a), and the cross section of its reflected field is shown in (b). The reflection and refraction of the Ey component in the x–z plane are shown in (c), and the cross section of its reflected field is shown in (d). The figure clearly shows that the reflected field is mostly polarized in the y direction. Note that the field values are normalized with respect to their respective incident fields.
E~ ix −I e kx kzk − I h ky μk ωeikx xky ykzk z ;
with the excitation coefficients
~ ix −I h kx kzk I e ky ϵk ωeikx xky ykzk z ; H E~ iy −I e ky kzk I h kx μk ωeikx xky ykzk z ; ~ iy −I h ky kzk − I e kx ϵk ωeikx xky ykzk z ; H
(12)
k ~ e ; I h − kzk Et− μk H t ∕D ~ h; I e − − zk H t− ϵk E t ∕D ω ω
1.5 Incident Reflected Refracted
1
kz/k
0.5 0 -0.5 -1
-1
-0.5
0
0.5
1
kx/k
(a)
(b)
Fig. 4. (a) Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kx –kz plane as given by Eq. (6). The support curve of the incident field is deformed such that the refracted field forms a Bessel beam with cone angle p ζ π∕6 and refraction angle ϑ π∕4 through a half-space with χ 2k. Note that for this cone angle and material, ϑ π∕4 is the maximum possible refraction angle for a complete Bessel beam, since greater angles would drive the incident spectrum into the evanescence region. (b) Bessel beam refracted at angle ϑ π∕6 into a medium with ϵχ 3.9ϵk and μχ μk . The incident and reflected fields do not form a Bessel beam; however, the incident field is converging at the interface.
M. A. Salem and H. Bağcı
~ e 2k2x k2y kzk ϵk and D ~ h 2k2x k2y kzk μk , and the where D refracted field spectra are those of the required Bessel beam. Figure 4 depicts the refraction of a full-vector Bessel beam at angle ϑ π∕6 due to an incident field determined by Eq. (12).
5. TOTAL REFLECTION AND SHIFT CASE Here, we consider the case where the refracted field is evanescent, or predominantly evanescent. Several conditions exist in which these criteria are met, resulting in total reflection for the whole or the greater part of the spectrum. In general, the necessary condition that must hold in order to have no beam-like propagating field in the refraction p medium is χ < k sin2 ζ cos2 ζ sin2 θ, where the media properties, the angle of incidence, θ, and the cone angle, ζ, are involved. In [12], the GH and IF shifts occurring upon normal incidence of a vector Bessel beam were qualitatively discussed in terms of the change in the transverse Poynting vector components. It was argued that the azimuthal component of the Poynting vector S ϕ transfers the energy along the beam ring and is responsible for the GH shift. The GH shift in this case manifests itself in the form of a beam rotation that cannot be visually seen. The radial component of the Poynting vector S ρ transfers the energy in the radial direction at the interface and is responsible for the IF shift. The IF shift in this case manifests itself in the form of a change in the beam spot size. The longitudinal component of the Poynting vector S z is equal to zero, since the evanescent Bessel beam does not carry energy along its axis.
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In the case of oblique incidence, two types of GH shifts take place; the first is a lateral shift along the direction of incidence (x axis), and the second is a rotational shift. To identify the lateral shift, we recall that for total reflection, the reflection coefficients of the field are complex with a magnitude equal to unity. Accordingly, if the reflection coefficients could be ~ expiγ, then the lateral GH is given written in the form G by Δx −dγ∕dkx . From Eq. (9), it is obvious that the two quantities De and Dh are the quantities responsible for the shifts. It also becomes evident that there are two distinct lateral shifts in the case of full-vector Bessel beam total reflection, which are given by Δxe
kx ϵk ϵχ k2 − χ 2 ; kzk jkzχ jϵ2χ k2zk − ϵk k2zχ
Δxh
kx μk μχ k2 − χ 2 : kzk jkzχ jμ2χ k2zk − μ2k k2zχ (13)
The excitation amplitudes and the refraction medium properties are thus the determining factors, of which shift is the dominant one. Additionally, for the field reflected from a right-handed material half-space, the phase shift is negative, resulting in a positive displacement along the x axis; whereas in the case of left-handed metamaterial half-space, a positive phase shift occurs and accordingly the field exhibits a negative displacement along the x axis. The refracted evanescent fields exhibit the same lateral shifts, except in the opposite direction. Quantification of the rotational GH shift and the IF shift is more involved, since both are related to phase change with respect to ky . We also note that in the case of oblique
Fig. 5. Comparison between the scattered fields upon incidence under total reflection condition. In (a), (b), and (c), the refraction medium is a right-handed material with ϵχ ϵk ∕3.9 and μχ μk , and in (d), (e), and (f), the refraction medium is a left-handed metamaterial with ϵχ −ϵk ∕3.9 and μχ −μk . In (a), a positive lateral GH shift along the x axis in the reflected Ex is visible in the x–z plane, in contrast to a negative GH lateral shift for the same component in (d). The lateral and rotational GH shifts for Ex are visible in (b) and (c), while these shifts are mirrored in (e) and (f). The change in the transverse field structure due to the IF shift is visible in (b), (c), (e), and (f).
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incidence, an “unspoiled” IF shift will not occur and the change in the spot size will suffer from unevenness. The reason for this unevenness is that the spectral support is symmetric with respect to the kx axis, but asymmetric with respect to the ky axis. The asymmetry across the ky axis results in a net displacement along the x axis (lateral GH shift), since the whole spectral support must be in the positive kx region to exhibit total reflection. On the other hand, the additional phase shift introduced by the reflection and refraction coefficients coupled by the symmetry of the spectrum support across the kx axis results in rotation (rotational GH shift) and change in the shape and size of the beam cross section (IF shift). Similarly, the phase change with respect to ky due to the quantities De and Dh is defined as Δy −dγ∕dky and is identical to Eq. (13) after exchanging kx ↔ky . Figure 5 depicts the shifts for oblique incidence upon two different half-spaces; the first is composed of a right-handed material, and the second is composed of a left-handed metamaterial. The figure clearly shows that the GH shifts in the case of the metamaterial half-space are reversed compared to the other case. The IF shift did not change, though, since it does not depend on the sign of the phase shift (direction of the displacement), but instead depends on its magnitude, which is the same for both cases.
6. CONCLUSION We have presented a pictorial illustrative method that describes the behavior of obliquely incident Bessel beams at a planar interface between two dielectric half-spaces with different material properties. This method is based on the algebraic spatial Fourier transform of the Bessel beam, where the spectral support of the Bessel beam is given by elliptic curves with complex weightings. Using this method, we deduced simple relations for the reflection and refraction of the field congruent to Snell’s laws. We also rigorously derived expressions for the scattered field after matching its tangential components at the interface. Final expressions of the scattered field in the spatial domain are determined by taking the inverse Fourier transform of the derived spectral field expressions. Using the obtained expressions of the scattered field, we deduced an expression for the polarization angle (Brewster’s angle) for obliquely incident Bessel beams and investigated the behavior of the scattered field upon incidence at such angle. We showed that the reflected field cannot be perfectly polarized due to the nature of the Bessel beam spectrum. Furthermore, we constructed a scheme to produce propagating Bessel beams in the refraction medium. This scheme requires that the incident field has a deformed spectral support curve and thus cannot be propagation-invariantlike Bessel beams. Finally, we have investigated the GH and IF shifts that occur when incident Bessel beams undergo total or almost total reflection. We have also shown that the GH shifts in the case of incidence upon a left-handed metamaterial half-space are the mirror images of the corresponding shifts if the half-space is composed of a right-handed material. The derived formulations and obtained results are principally important in fundamental understanding of the interaction of Bessel beams at planar interfaces, and applications that involve their scattering. The presented
M. A. Salem and H. Bağcı
simplified formulations of the refraction angle, the polarization angle, and the qualitative description of the scattered field behavior based on its spectral support provide practical and quick tools for understanding and engineering the interaction of Bessel beams. On the other hand, the rigorous formulation of the scattered field in the Fourier space provides a computationally inexpensive method to accurately determine the exact field in space. The work presented here lends itself to further extensions. Such extensions include investigating the behavior of obliquely incident vector Bessel beams on dielectric slabs and multilayer structures. This is particularly of interest in practical generation of laser Bessel beams, since gas laser devices rely on Brewster windows, which essentially are dielectric slabs with finite extension. Also the interaction of Bessel beams with biological tissues could be modeled as interaction with multilayer structures. This work could also be extended to pulsed Bessel beams and X waves, which are a chromatic collection of Bessel beams with extra conditions governing the relation between their spatiotemporal spectra.
ACKNOWLEDGMENTS This work was supported in part by the Center for Uncertainty Quantification in Computational Science and Engineering at King Abdullah University of Science and Engineering (KAUST).
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