Transmission and reflection of vector Bessel beams through an interface between dielectrics

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Transmission and reflection of vector Bessel beams through an interface between dielectrics

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Instituto de Física, Universidade Federal de Alagoas, Maceió – AL 57072-970, Brazil

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Article history: Received 11 November 2016 Received in revised form 26 December 2016 Accepted 27 December 2016 Available online xxxx Communicated by V.A. Markel Keywords: Bessel beam Orbital angular momentum Total internal reflection Brewster angle

Simple full vectorial analytical exact results are obtained for the propagation of Bessel beams through an interface separating different media characterized by material parameters (1 , μ1 ) and (2 , μ2 ). A real space description is used and all the results are written in terms of the transverse wavevector kt of the incident beam, taken as an input parameter. It is shown that perfect transmissions are obtained when the incident wave is either in a medium of larger or lower index of refraction, compared to the medium of the transmitted wave, and we provide the particular values, ktF , that the transverse wavevector must obey in order to observe such effects. The phenomenon of total internal reflection is also verified for normal incidence. © 2016 Published by Elsevier B.V.

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1. Introduction

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One of the main concerns of physical optics is to understand the classical interaction and propagation of arbitrary waves through general materials. By considering the propagation of fully linearly polarized waves, Fourier optics adequately provides an explanation for various effects by treating the electromagnetic field as a scalar quantity [1]. Nevertheless the electromagnetic field is characterized by a vector field and must obey Maxwell’s equations, so that any consideration about the polarization of the waves in question cannot be taken into account by using a pure scalar approximation. In 1987, Durnin discovered such a scalar solution of the Helmholtz equation, namely, the Bessel beam [2]. The search began for providing a vectorial formal solution of Maxwell’s equations and at the same time trying to preserve essential features of Durnin’s solution, such as the nondiffractive character and orbital angular momentum properties [3–5]. Unfortunately, there is no general approach to this problem and the interaction of Bessel beams (also known as X waves) with ordinary matter is studied with several distinct formalisms. Phenomena such as superluminal propagation properties through slabs [6,7], total internal reflection [8], changes on the size of the Bessel beam rings under reflection [9], interaction with absorbing media [10,11] are well established in the literature. We cite in particular the more general formalism developed in [12,13] which closely resembles ours. The importance of understanding these solution stems from the

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E-mail address: [email protected] (P.A. Brandão). http://dx.doi.org/10.1016/j.physleta.2016.12.049 0375-9601/© 2016 Published by Elsevier B.V.

remarkable range of important applications that Bessel beams can provide, in particular, optical micromanipulation of microsized particles [14], three dimensional imaging of living cells [15] and optical levitation [16] to cite a few. Once vectorial solutions are obtained, the next logical step is to make the interaction of the beam with some material. In a previous paper [5] we have obtained a vectorial Bessel beam solution of Maxwell’s equations and it was shown that it possesses properties resembling Durnin’s solution. In this paper we take a step forward in asking what happens to this vector beam as it interacts with an interface separating dielectric materials characterized by material parameters (1 , μ1 ) and (2 , μ2 ) taken to be real positive quantities. In particular, we have found special configurations in this system such that total internal reflection and total transmission are obtained even when the Bessel beam undergoes normal incidence.

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2. Theory

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The analysis begins by presenting full vectorial analytic solutions of Maxwell’s equations for two types of polarization modes TE and TM. The derivation was established in a previous paper [5] and therefore only the essential results will be presented in the following. By considering time harmonic propagation exp(−i ωt ) and an electric field and magnetic field induction of the form E(x, y , z, t ) = E(x, y ) exp(ik z z − i ωt ) = [E⊥ (x, y ) + E z (x, y )ˆz] exp(ik z z − i ωt ) and B(x, y , z, t ) = B(x, y ) exp(ik z z − i ωt ) = [B⊥ (x, y ) + B z (x, y )ˆz] exp(ik z z − i ωt ), where z is the propagation direction, ω the angular frequency, k z the wavevector in the z

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P.A. Brandão, D.G. Pires / Physics Letters A ••• (••••) •••–•••

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direction and E⊥ (x, y ) = zˆ × [E(x, y ) × zˆ ] (same for B⊥ ), which is the transverse part of the electric field (ˆz · Et = 0), it can be demonstrated that



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E(x, y , z, t ) = exp(ik z z − i ωt ) −



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B(x, y , z, t ) = exp(ik z z − i ωt )

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iω 2

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for TE polarization, where k⊥ = μω where  and μ are the (position independent) dielectric constant and permeability of the medium where the optical beam propagates, respectively. The field 2 component B z (x, y ) must be a solution of (∇⊥ + k2⊥ ) B z (x, y ) = 0 2 where ∇⊥ = ∇ 2 − ∂z2 . Transforming to cylindrical coordinates one may choose B z (ρ , φ) = B 0 J m (k⊥ ρ ) exp(imφ) where B 0 is the amplitude (taken to be real), J m (k⊥ ρ ) is the m-order Bessel function of the first kind and m = 0, 1, 2... denotes the order of the solution. By substituting B z in Eqs. (1) and (2), the following equations for TE polarized Bessel beam are obtained 2

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E(ρ , φ, z, t ) = E 0 exp(ik z z + imφ − i ωt ) φˆ J − − i ρˆ J + ,

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[ρˆ J − + i φˆ J + ] + J m (k⊥ ρ )ˆz ,

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where J ± = J m−1 (k⊥ ρ ) ± J m+1 (k⊥ ρ ) and E 0 = −i ω B 0 /2k⊥ . The TM solutions can be obtained quite easily by following the preceding analysis and we will study them later. By doing a simple calculus problem it can be demonstrated that ∇ · E = 0 and ∇ · B = 0 as required by Maxwell’s equations without sources. With these full vectorial solutions one is ready to impose electromagnetic boundary conditions.

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3. Applications and discussion

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Fig. 1. Diagram showing the radius r1 = (1 μ1 ) ω and r2 = (2 μ2 ) ω for the case r2 > r1 . For the region (a) it is shown propagating waves on both sides of the structure. The region outside the white circle is forbidden for propagating incident waves. The point (a) indicates a value k⊥ for the incident wave. 1/2

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To this end, consider that the half-space z ≥ 0 is occupied by a material characterized by (2 , μ2 ), z ≤ 0 by a material characterized by (1 , μ1 ) and a Bessel beam of order m is normally incident from negative to positive z. After using the usual Maxwell’s equations boundary conditions at z = 0 one may deduce the following relations:

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where B 0 ( E 0 ), B 0 ( E 0 ) and B 0 ( E 0 ) are the amplitudes of the incident, reflected and transmitted magnetic (electric) fields, respectively, k z = (1 μ1 ω2 − k2⊥ )1/2 and kz = (2 μ2 ω2 − k2⊥ )1/2 are the z components of the wavevector for z < 0 and z > 0. In deriving Eqs. (5) and (6) the continuity of the transversal wavevector was used. We also assumed that initially all three, i.e., the incident, reflected and transmitted waves, had different values of m, but after substitution into the original equations for the fields, it can be demonstrated that m = m = m , which shows that the reflected and transmitted beams must have the same order as the incident one. All the relevant equations can be written in terms of the single transverse wavevector k⊥ parameter of the incident beam (for a given set of material parameters and angular frequency ω ). As we are concerned with propagating incident waves, k⊥ must always be smaller than (1 μ1 )1/2 ω . Now, either 1 μ1 < 2 μ2 or 1 μ1 > 2 μ2 . For the first condition (a Bessel beam traveling from air to glass, for instance) let r1 = (1 μ1 )1/2 ω and r2 = (2 μ2 )1/2 ω

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which corresponds to the radius shown in Fig. 1. As k⊥ must always be smaller than r1 for an incident propagating wave (which corresponds to the white region in Fig. 1), k⊥ < r2 and, consequently, kz is real and the transmitted wave propagates without decaying. Geometrically, it is clear that for every point inside the white circle of radius r1 it will also be inside of the larger circle with radius r2 . We conclude that it is impossible to have evanescent behavior for z > 0 if 1 μ1 < 2 μ2 . Now, for a single TE polarized plane wave incident onto the interface, there is an angle, called Brewster’s angle, for which there is no reflected wave [9]. Making the analogy with our system, we ask if there is such behavior for the vector Bessel beam. By looking at Eq. (5) it is seen that if μ2 k z = μ1 kz the amplitudes of the reflected electric and magnetic waves are zero. This happens for a k⊥ given by



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Given (1,2 , μ1,2 ) parameters, the transverse wavevector for full transmission can be calculated using Eq. (7). Be aware that the F condition k⊥ < r1 must also be obeyed. We will give examples later showing that the conditions can be fulfilled, though. Note that if the frequency is in the optical spectrum where μ1 ≈ μ2 ≈ μ0 , this effect of full transmission probably will not be observable at F all, for k⊥ becomes much larger than r1 . We conclude that an analogous Brewster’s angle exists for this situation but it is a little more subtle than the plane wave solution. Referring to Fig. 1 it may be said that there are some points inside the circle with radius r1 for which there is a full transmission of the incident beam. Later we will demonstrate a specific example where these points can be visualized. Consider now 1 μ1 > 2 μ2 (r1 > r2 ) which can be thought of as a Bessel beam propagating from glass to air, for instance. The diagram representing r1 and r2 is shown in Fig. 2. For this situation, k⊥ still can have values lying inside both circles, as represented by the green shaded area. This represents propagating waves in both materials. But now there are points satisfying r2 < k⊥ < r1 , such as point (a) in Fig. 2. For these points kz becomes pure imaginary and an evanescent wave appears in the region z > 0. This is analogous to the total internal reflection phenomenon of a polarized plane wave incident upon a material from a higher to a lower index of refraction. What plays the role of the critical angle here C is the critical wavevector k⊥ = ω(2 μ2 )1/2 , i.e., when the incident transverse wavevector matches the value of r2 . We conclude that as k⊥ acquires large values and becomes closer to r2 , the state of the transmitted electromagnetic field goes through a transformation from a propagating to an evanescent state. It must be pointed out that Eq. (7) can still be satisfied for the green shaded area of F Fig. 2 (this time we must have k⊥ < r2 ). Full transmission of the incident wave through the boundary from the medium with larger refraction index to the lower (and vice versa) is possible for the vectorial Bessel beam.

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Fig. 2. Diagram showing the radius r1 = (1 μ1 ) case r2 < r1 .

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Fig. 4. Time averaged energy density for r1 < r2 and k⊥ satisfying Eq. (7). The electromagnetic wave is totally transmitted through the material without reflections.

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Fig. 3. Time averaged energy density for r1 < r2 and k⊥ = r1 /2. It can be seen the pattern of interference for z < 0 and the transmitted wave for z > 0.

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In order to have theoretical results connected with any experiment that involves capturing the intensity pattern in a CCD (charge coupled device camera) for example, we calculated the time averaged energy density given by u = (1/4)[ E · E∗ + (1/μ)B · B∗ ] for z < 0 and z > 0. In all calculations the value m = 3 was considered. Suitable material parameters μ1 = μ2 = μ0 , 1 = 0 and 2 = 100 are chosen such that r1 = ω(1 μ1 )1/2 < r2 . The situation is as illustrated in Fig. 1. The energy density has azimuthal symmetry and the distribution in the plane y = 0 is considered. It can be seen from Fig. 3 that for z < 0 an interference pattern between the incident and reflected waves is formed, and for z > 0 it is shown the transmitted wave. The vertical gray line in Fig. 3 indicates the boundary between the two materials (z = 0 plane) and the colorbar at the right merely indicates that red corresponds to the maximum value of the energy density and blue corresponds to the minimum (same for the other figures). Let us now take material parameters such that Eq. (7) is satisfied. For this we take 1 = 0 , μ1 = μ0 , 2 = 40 and μ2 = 5μ0 . Still r1 < r2 but k⊥ is as Eq. (7). In Fig. 4 it is shown the energy density results for this particular situation. It is easy to see that the Bessel beam is fully transmitted through the boundary without any reflections. This is the analog of the Brewster angle effect for vector Bessel beams under normal incidence. We should emphasize that by saying normal incidence we are indicating that there exists a more general incidence profile than that of the Figs. 3 and 4. This happens when, for example, the horizontal intensity lines in Figure 4 are inclined with respect to the z direction. We did not consider this general situation in this paper. For the last example consider r1 > r2 and the point (a) in Fig. 2. By choosing 1 = 100 , μ1 = μ2 = μ0 , 2 = 0 and k⊥ = r1 /2 the relations are satisfied. The wave for z > 0 must be evanescent and that is shown in Fig. 5. The spatial decaying constant is easily shown to be given by γ −1 = (k2⊥ − 2 μ2 ω2 )1/2 . For TM polarization, the electric and magnetic fields are given by

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Fig. 5. Time averaged energy density for r1 > r2 and k⊥ = r1 /2. The electromagnetic wave is evanescent for z > 0 corresponding to the region of the point (a) in Fig. 2. The positive part of z was scaled to 1/8.

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where B 0 = μ E 0 /2k⊥ . By doing the same boundary value problem as before, it can be demonstrated that the relations between reflected and transmitted electromagnetic amplitudes with the incident ones are given by

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By looking at Eq. (12) it can be noticed that the total transmission for TM polarization should be more pronounced in the optical spectrum, for we no longer have the μ1 = μ2 restriction. We believe that the physical reason responsible for the full transmission in the optical spectrum for TM, as opposed to TE polarization, is that for TM polarized beams the z component of the electric field induces a surface bounded charge given by

σ B = 20 E 0 J m (k⊥ ρ ) cos(mφ − ωt )

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Considering total transmission through the boundary (zero reflected field amplitudes), the wavevector for fully transmitted beam is now given by



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2k z − 1kz , = = E0 B0 2k z + 1kz   E 0 1 μ1 B 0 22 k z . = = E0 2 μ2 B 0 2k z + 1kz



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χ 2 − χ1 (1 + χ2 ) + (1 + χ

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which can be calculated by using the boundary conditions for the normal components of the electric fields, where 0 is the free space dielectric permittivity and χi the linear optical susceptibility of the (i , μi ) medium. These bounded electrons should respond more strongly to the incident fields, making possible the full transmission for the optical spectrum (the TE polarization mode creates no electronic surface charges). Further work must be performed to understand this difference, though. In some cases it is possible to have a pictorial visualization of the region inside the circle with radius r1 for which full transmissions are possible. For this, consider frequencies in the optical

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in which nondiffracting and self-healing properties are maintained and, under this range, a Bessel-like beam could be able to reproduce our theoretical results with good agreement. The generation of Bessel-like beams and its interaction with ordinary matter are still in early developments, but experimental conditions are advancing at a rapid pace and we hope our results could be physically achievable in the near future.

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4. Conclusions

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Fig. 6. The quantity k⊥ /r1 = f is plotted as a function of 2 /0 = w. For values of 2 higher than 0 the wave is fully transmitted through the boundary. The horizontal √ dashed line represents the line f = 1/ 2. F

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In conclusion, new effects arise when one considers the simple problem of a non-plane wave incident on a boundary surface. Under normal incidence Bessel beams can suffer total transmission as well as total internal reflections (which plane wave solutions can only undergo by oblique incidence). A new behavior and interpretation must appear from mixed polarization as well as inclusion of more boundary surfaces, a problem still waiting for a solution. The circle diagrams used in Figs. 1, 2 and 7 may show some value as more variables are added into the problem and this will be treated elsewhere. The present results are important for any application that involves propagation of general vector beams through a dielectric cascade which includes a wide scientific audience.

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Acknowledgements

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The authors would like to thank the Brazilian Agencies CAPES and CNPq for financial support and S.B. Cavalcanti for a critical reading of the manuscript.

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Fig. 7. Pictorial representation of the points inside the circle with radius r1 for which full transmission is possible. For this figure, 1 = 0 , r1 = ω(0 μ0 )1/2 , r2 = ω(2 μ0 )1/2 and r3 = r1 [(2 /0 )/(2 /0 + 1)]1/2 . The blue region corresponds to points of 2 such that the field is fully transmitted. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

spectrum where it is reasonable to put μ1 = μ2 = μ0 such that F Eq. (12) reduces to k⊥ = ω[μ0 1 2 /(2 + 1 )]1/2 = r1 r2 /(r12 + r22 )1/2 . Consider the special case 1 = 0 , so r1 = ω(0 μ0 )1/2 and we must F F choose 2 such that k⊥ < r1 . Fig. 6 shows the plot of k⊥ /r1 = f as a function of 2 /√ 0 = w for 1 = 0 . From this figure, a minimum F value of k⊥ = r1 / 2 exists and this represents the inner radius of the blue region in Fig. 7. By increasing 2 , the maximum value F k⊥ = r1 is reached. Thus, Fig. 7 shows explicitly the points of Fig. 1 from which full transmission is possible for this particular set of parameters. A remark should be made about the physical possibility of actually creating such complex vector fields represented by Bessel functions in a laboratory. It is important to notice that Bessel beams cannot be created in its full characteristics because they require an infinite amount of energy [2,5]. On the other hand, Bessellike beams are able to be experimentally generated by means of a spatial light modulator (SLM) [18] or by using an axicon associated with a Mach–Zehnder interferometer [19], to cite two methods. In this physical situation there is a maximum propagation range

Uncited references

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