November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS

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Cylindrical quasi-Gaussian beams F. G. Mitri Los Alamos National Laboratory, MS D429, Los Alamos, New Mexico 87545, USA ([email protected]) Received September 13, 2013; accepted October 8, 2013; posted October 10, 2013 (Doc. ID 197608); published November 12, 2013 Making use of the complex-source-point method in cylindrical coordinates, an exact solution representing a cylindrical quasi-Gaussian beam of arbitrary waist w0 satisfying both the Helmholtz and Maxwell’s equations is introduced. The Cartesian components of the electromagnetic field are derived stemming from different polarizations of the magnetic and electric vector potentials based on Maxwell’s vectorial equations and Lorenz’s gauge condition, without any approximations. Computations illustrate the theory for tightly focused and quasi-collimated cylindrical beams. The results are particularly useful in beam-forming design using high-aperture or collimated cylindrical laser beams in imaging microscopy, particle manipulation, optical tweezers, and the study of scattering, radiation forces, and torque on cylindrical structures. © 2013 Optical Society of America OCIS codes: (260.1960) Diffraction theory; (260.2110) Electromagnetic optics. http://dx.doi.org/10.1364/OL.38.004727

Focused cylindrical wave-fields, which can be tailored to produce magnification of the energy in only one plane and provide better system efficiency, have received significant attention [1–5] in imaging applications and microscopy, particle trapping, electron acceleration, and optical data storage, to name a few applications. Generally, the mathematical description of a scalar field corresponding to a monopole oscillating source in a cylindrical coordinate system ρ; θ; z with an outwardly symmetric cylindrical radiation wave-front tilted by an angle of incidence α with respect to the z axis is expressed by a zero-order cylindrical Hankel function of the first kind H 1 0 ⋅ as ikz z−ωt ; ψ  ψ 0 H 1 0 kρ ρe

(1)

where ψ 0 is the characteristic amplitude, kρ  k cos α, kz  k sin α, k is the wavenumber, ρ is the radial distance in the polar plane x; y defined by ρ  x − x0 2  y − y0 2 1∕2 , where x0 and y0 are the source coordinates, and the exponential e−iωt defines the time dependence. As noticed from Eq. (1), a singularity arises at the center of the coordinate system ρ  0. If a complex coordinate system is used such that x0  ixR and y0  0, where i is the imaginary unit, the wave function that is due to a source placed at the imaginary distance x0 from the origin is an exact solution of the wave equation. Therefore, Eq. (1) becomes ikz z−ωt ψ ∓  ψ 0 H 1 ; (2) 0 kρ ρ∓ e p where the complex distance, ρ∓  x∓ixR 2  y2 , is a multiple-valued (in this case four-valued) function [6] in the real x; y plane. The effect of describing the cylindrical wave solution with a complex argument kρ ρ∓ , which may appear at first glance a simple artifice, has a major physical meaning in the description of evanescent waves [7] and the production of directional (focused or collimated) beams [8] with a waist w0 , using the complex-source-point (CSP) method [6,9–13]. Actually, near the x axis the ∓ solution in Eq. (2) behaves like a Gaussian beam that propagates along the ∕ − x axis, respectively. The parameter xR is identified as the Rayleigh length of these beams; hence it is related to the waist w0 via

0146-9592/13/224727-04$15.00/0

xR  kw20 ∕2:

(3)

As the diffraction limit is reached when w0 ≈ λ∕2 [14], where λ is the wavelength of the incident radiation, depending on the value of the dimensionless waist kw0 , the cylindrical beam can be tightly focused (or strongly divergent) (i.e., kw0 < 3), or quasi-collimated (i.e., kw0 > 3) in the broad sense. In order to make ρ∓ single-valued, it is necessary to introduce branch lines (or cuts) in the x; y plane, which emerge from the branch points at x  xR and x  −xR . The choice of the branch lines affects the properties of ρ∓ (Figs. 2(a) and 2(b) in [6]), so this function is continuous at all points of the region except the branch cuts. Although Eq. (2) is an exact solution of the Helmholtz equation, a problem with the CSP method remains; that is, Eq. (2) possesses a singularity at the CSP ρ∓  0. The branch singularity occurs in the plane of the waist, making it an inappropriate solution for applications where one is interested specifically in the field at that particular plane. Fortunately, this singularity can be eliminated by adding a sink [15,16] to the point source with the same amplitude and opposite sign. Thus, the physically realizable solution is expressed as ψ∓ r 

ψ 0 1 ikz z−ωt H 0 kρ ρ∓  − H 1 : 0 −kρ ρ∓ e 2

(4)

Using the identity for the cylindrical Hankel functions of the first and second kind of order ν, H 1 ν −kρ ρ∓   −1ν1 H 2 ν kρ ρ∓  when jkρ ρ∓ j > 0, Eq. (4) can be expressed in terms of the zero-order cylindrical Bessel function J 0 ⋅ as ikz z−ωt ψ∓ : r  ψ 0 J 0 kρ ρ∓ e

(5)

Unlike Eq. (2), the solution in Eq. (5) is free from any singularity, is finite at the CSP ρ∓  0, and remains an exact solution of the Helmholtz equation. The cost of this regularization, however, is that the solution in Eq. (5), which behaves like a Gaussian beam that propagates along the ∕ − x, respectively, theoretically contains a weak field component propagating backwards in the −∕  x direction, respectively, with respect to the main beam [15]. It is termed here “quasi-Gaussian” to make a clear distinction from the paraxial Gaussian solution © 2013 Optical Society of America

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OPTICS LETTERS / Vol. 38, No. 22 / November 15, 2013

that does not satisfy the Helmholtz equation. Moreover, in the real source case, Eq. (5) corresponds to the mathematical description of a scalar Bessel beam of order zero [17,18] propagating along the z direction. With a large beam waist, computations of the modulus of the wave function given by Eq. (5) show that the beam’s amplitude (or intensity) varies slowly along the direction of wave propagation; consequently, the beam may be described using the theory of scalar fields. However, in the case of strongly focused beams (i.e., a small beam waist), it is necessary to account for the vector nature of the electromagnetic (EM) radiation. The procedure requires using the dual-field setup [19] to derive the field’s components in free space with complex Cartesian coordinates. This process has also been applied in the description of vector Bessel beams [20–25], and spherical quasi-Gaussian beams [26]. Assuming that the incident beam propagates along the negative x direction, a magnetic vector potential field polarized along the y direction is considered such that Ψm;y 

ψ 0 J 0 kρ ρ eikz z−ωt y: I 0 kρ xR 

(6)

Note that since the beam propagates along the x axis, the polarization of the vector potential is adequately chosen to be in the transversal plane (i.e., in the y or z directions), so that near the beam’s axis the field has a quasi-TEM form [27] [see also Eqs. (18)–(23) below]. One may alternatively consider the solution with ρ− (which propagates along the positive x direction); however, it can be proved that the end results will be the same. Since Eq. (6) is expressed in terms of the cylindrical Bessel functions of zeroth order, it represents the lowest-order solution of quasi-Gaussian cylindrical beams. The factor in the denominator is adjusted so as to obtain Ψm;y  ψ 0 y at the center of the cylindrical coordinate system ρ; z  0; 0, and I 0 · is the modified cylindrical Bessel function of the first kind of order zero. Using Lorenz’s gauge condition [28], a magnetic field Hm can be defined as −1∕2

∇ × Ψm;y  Hm ε

;

Ψe;y  −

(8)

The Cartesian components for the magnetic and electric fields may be deduced from Eqs. (7) and (8), respectively. However, an asymmetry arises in the resulting equations, which is unsuitable in the general (i.e., with no imposed boundary conditions) mathematical representation of symmetric beams. [Note since Ψm;y is polarized along the y direction, Eq. (7) indicates that the transverse component H m;y  0]. To produce symmetrical behaviors in the magnetic and electric fields, it is necessary to use the dual-field setup [19], and define an appropriate electric vector potential field (to be added to the solution stemming from the magnetic vector potential field) taking into account the geometry of the problem. One may choose an electric vector potential

ψ 0 J 0 kρ ρ eikz z−ωt y: I 0 kρ xR 

(9)

Consequently, an electric field Ee is defined as ∇ × Ψe;y  Ee :

(10)

From Maxwell’s equations and Eq. (10), the magnetic field is expressed as He ε−1∕2  −ikΨe;y  ∇∇ · Ψe;y ∕k2 :

(11)

Adding the solutions of Eq. (7) with Eq. (11), and Eq. (8) with Eq. (10), and dividing the end result by two, leads to the spatial Cartesian components for a cylindrical quasi-Gaussian beam, which are expressed as E y;−y x

  iψ 0 eikz z−ωt ρ kkz ρ2 − k2ρ yX  J 0 kρ ρ   ; 2kρ3 I 0 kρ xR  2kρ yX  J 1 kρ ρ  (12)

 E y;−y y

iψ 0 eikz z−ωt 2kρ3 I 0 kρ xR    ρ kρ − kρ ykρ  kρ yJ 0 kρ ρ  ; (13) × −kρ ρ2 − 2y2 J 1 kρ ρ 

E zy;−y 

H y;−y x

ψ 0 eikz z−ωt k J k ρ X   y sin α; 2ρ I 0 kρ xR  ρ 1 ρ 

(14)

 p  iψ 0 eikz z−ωt ε ρ kkz ρ2  k2ρ yX  J 0 kρ ρ  ; − 2kρ3 I 0 kρ xR  −2kρ yX  J 1 kρ ρ  (15)

(7)

where ε is the dielectric constant of the medium. Thus, from Maxwell’s equations and Eq. (7), the electric field is expressed as Em  ikΨm;y  ∇∇ · Ψm;y ∕k2 :

polarized along the negative (or positive) z direction; however, this particular choice still produced asymmetry between the components. Thus, another suitable polarization for the electric vector potential Ψe;y is sought along the negative (or positive) y direction, such that

H yy;−y  H y;−y z

p y;−y εE y ;

(16)

p ψ 0 eikz z−ωt ε k J k ρ X  − y sin α; − 2ρ I 0 kρ xR  ρ 1 ρ  (17)

where the superscript y; −y in Eqs. (12)–(17) denotes the polarization state of the vector potentials Ψm;y and Ψe;y , respectively. Moreover, the function appearing in Eqs. (12), (14), (15), and (17), which represents the complex axial distance, is expressed as X   x  ixR . Repeating the derivations for another choice of vector potentials polarized along the transversal z directions, the EM field’s components are found to be E xz;−z 

ψ 0 eikz z−ωt k J k ρ y  X  sin α; 2ρ I 0 kρ xR  ρ 1 ρ   ε−2 H zy;−y ; E z;−z y 1

(18) (19)

November 15, 2013 / Vol. 38, No. 22 / OPTICS LETTERS

E zz;−z  H xz;−z  −

iψ 0 eikz z−ωt k J k ρ  cos α; 2I 0 kρ xR  ρ 1 ρ 

(20)

p ψ 0 eikz z−ωt ε k J k ρ y − X  sin α; (21) 2ρ I 0 kρ xR  ρ 1 ρ  H yz;−z 

p y;−y εE z ;

(22)

H zz;−z 

p z;−z εE z :

(23)

Now changing the polarization states of the vector potentials with the assumption that the magnetic (or electric) vector potential is polarized along the direction of wave propagation (or opposed to it) produces components quite distinct from those obtained previously. This had been suggested in the development of a new type of free electron lasers [29]. Considering a magnetic vector potential Ψm;x polarized along the positive direction x, and another electric vector potential Ψe;x polarized along the opposite direction −x, and reiterating the arithmetic steps based on the procedure described previously, leads to the derivation of the spatial components of the EM field in the x; −x configuration as iψ 0 eikz z−ωt 2kρ3 I 0 kρ xR    ρ kρ − kρ X  kρ  kρ X  J 0 kρ ρ  × ; (24) −kρ ρ2 − 2X 2 J 1 kρ ρ 

Ex;−x  x

 Ex;−x y

  ψ 0 eikz z−ωt 2ikρ X  yJ 1 kρ ρ  ; 2 2 2kρ3 I 0 kρ xR  −ρ ikkz ρ  ikρ X  yJ 0 kρ ρ  (25)

− E x;−x z

To illustrate the theoretical analysis, plots of the moduli of the EM field’s components and the EM power density in the transversal y; −y, z; −z and the axial x; −x directions are computed in the transverse x; y-plane for the cases of interest corresponding to a strongly focused (kw0  0.1) and a quasi-collimated (kw0  4) cylindrical quasi-Gaussian beam with emphasis on the tilt angle α. In the simulations, the parameter k  25 × 103 m−1 , and the x; y coordinates are varied by increments of δx; y; z  10−3 mm. Figure 1 shows the plots for a tightly focused cylindrical quasi-Gaussian beam with kw0  0.1 and α  0°. The similarity of the plots for the EM field’s components in each configuration [i.e., transverse (a), (b) and axial (c)] is apparent. This is not the case for the power density; in the axial polarization [lower panels in Fig. 1(c)], it vanishes along the direction of the beam propagation x. Comparisons of the panels in Fig. 1 with those of Fig. 2 show the effect of changing the tilt angle α  45°, where some similarities remain in the plots of Fig. 2. Nevertheless, one notices the asymmetry with respect to the y direction that arises by changing the tilt angle α. Moreover, the power density plot shows a central null for the transverse polarization z; −z [lower panels in Fig. 1(b)], and the axial null, noted in the lower panels of Fig. 1(c), is transformed into a peak at the center of the transverse plane x; y  0; 0. In addition, side-lobes appear in the power density plots [lower one-line profile plots, Figs. 1(a)–1(c)], which were not observed for the case where α  0° [lower panels in Figs. 1(a) and 1(b)]. The effect of varying the dimensionless waist kw0  4 for a tilted beam (with α  45°) is shown in Fig. 3. In contrast to Fig. 2, the quasi-collimation characteristic is clearly displayed as the focal area becomes elongated in space for all components. A spatial asymmetric distribution is also observed, which is due to α ≠ 0. Furthermore, the side-lobes around the main focal area

ψ 0 eikz z−ωt k y − X  sin αJ 1 kρ ρ ; (26) 2ρ I 0 kρ xR  ρ  H x;−x x

 H x;−x y

4729

p x;−x εE x ;

(27)

 p  ψ 0 eikz z−ωt ε 2ikρ X  yJ 1 kρ ρ  ; 2 2 2kρ3 I 0 kρ xR  ρ ikkz ρ −ikρ X  yJ 0 kρ ρ  (28)

H x;−x z

p ψ 0 eikz z−ωt ε k y  X  sin αJ 1 kρ ρ : (29)  2ρ I 0 kρ xR  ρ

It is important to note here that all the EM field’s components derived based on the chosen polarizations [i.e., Eqs. (12)–(29)] are symmetric among one another. Moreover, those are essential in the numerical predictions of the arbitrary scattering [30,31], radiation forces and torque induced by such quasi-Gaussian beams. Another observable of interest in focused cylindrical lasers is the EM power density, expressed as [28] hSi  c∕8πReE × H :

(30)

Fig. 1. Planar cross-sectional plots of the modulus of the electric and magnetic fields’ components, in the transverse y; −y, z; −z and axial x; −x configurations, denoted by (a)–(c), respectively, for a strongly focused beam with kw0  0.1 and α  0. The lower panels display the cross-sectional plots for the power density with their one-line profiles at the center of the beam. The units along the axes are in millimeters.

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θ∓  cos−1 x∓ixR ∕ρ∓ :

Fig. 2.

Same as in Fig. 1, but α  45°.

that were manifested in the power density plots in Fig. 2 have been all smoothed out. In all configurations, the power density is slightly affected as it remains well confined into a quasi-collimated area. The aim of the present investigation was to introduce the scalar solution given by Eq. (5) [or the vectorial one given by Eq. (6)] to describe the lowest-order mode of cylindrically focused or (quasi-)collimated quasiGaussian beams. The same analysis can be extended to include scalar (or vectorial) higher-order cylindrical quasi-Gaussian beam solutions to the Helmholtz and Maxwell’s equations. The results expressed in terms of scalar higher-order CSP cylindrical waves are given by ikz zlθ∓ −ωt ;  ψ l ψ l∓ r 0 J l kρ ρ∓ e

(31)

or alternatively, ikz z−ωt  ψ l coslθ∓ ; ψ l∓ r;t 0 J l kρ ρ∓ e

(32)

where l ≠ 0 is a real integer defined as the order of the beam, and θ∓ is the complex angle defined as

Fig. 3. Same as in Fig. 1, but kw0  4 and α  45°.

(33)

Preliminary computations (not displayed here) have shown that the resulting moduli plots of both the scalar functions given by Eqs. (31) and (32) are quite distinct, and their corresponding beam profiles are different. There also exist numerous combinations that produce exact solutions of the Helmholtz equation; for example, adding a scalar beam function in the form of Eq. (31) with a positive order l to a second beam function of different amplitude and with negative order −l produces a unique solution with properties yet to be revealed. Another combination may consider adding the positive l− ψ l (or ψ l (or ψ l− r r;t ) to the negative ψ r r;t ) solution to produce another result satisfying the Helmholtz equation. The investigation of such combinations (or others) may be useful in beam-forming design from the standpoint of diffraction theory and EM optics, and possibly other applications. Further investigations, which are beyond the scope of the present analysis, are needed to fully analyze the properties of the generalized higherorder solutions and their superposition. References 1. J. S. Marsh, Am. J. Phys. 52, 152 (1984). 2. K. Youngworth and T. Brown, Opt. Express 7, 77 (2000). 3. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003). 4. Y. Kozawa and S. Sato, Opt. Express 18, 10828 (2010). 5. C. J. R. Sheppard, Appl. Opt. 52, 538 (2013). 6. E. Heyman and L. B. Felsen, J. Opt. Soc. Am. A 18, 1588 (2001). 7. L. B. Felsen, J. Opt. Soc. Am. 66, 751 (1976). 8. L. B. Felsen, Geophys. J. R. Astr. Soc. 79, 77 (1984). 9. Y. A. Kravtsov, Radiophys. Quantum Electron. 10, 719 (1967). 10. G. A. Deschamps, Electron. Lett. 7, 684 (1971). 11. J. B. Keller and W. Streifer, J. Opt. Soc. Am. 61, 40 (1971). 12. M. Couture and P. A. Belanger, Phys. Rev. A 24, 355 (1981). 13. C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998). 14. A. E. Siegman, Lasers (University Science Books, 1986). 15. E. Heyman, B. Z. Steinberg, and L. B. Felsen, J. Opt. Soc. Am. A 4, 2081 (1987). 16. M. V. Berry, J. Phys. A 27, L391 (1994). 17. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987). 18. J. Durnin, J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987). 19. A. L. Cullen and P. K. Yu, Proc. R. Soc. London 366, 155 (1979). 20. F. G. Mitri, Opt. Lett. 36, 606 (2011). 21. F. G. Mitri, Opt. Lett. 38, 615 (2013). 22. F. G. Mitri, Phys. Rev. A 85, 025801 (2012). 23. F. G. Mitri, Optik 124, 1469 (2013). 24. F. G. Mitri, Eur. Phys. J. D 67, 1 (2013). 25. F. G. Mitri, Phys. Rev. A 88, 035804 (2013). 26. F. G. Mitri, Phys. Rev. A 87, 035804 (2013). 27. E. Heyman, B. Z. Steinberg, and R. Iancunescu, IEEE Trans. Antennas Propag. 38, 957 (1990). 28. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999). 29. L. W. Davis and G. Patsakos, Opt. Lett. 6, 22 (1981). 30. J. P. Barton, J. Opt. Soc. Am. A 16, 160 (1999). 31. G. Gouesbet and G. Gréhan, Generalized Lorenz-Mie Theories, 1st ed. (Springer, 2011).

Cylindrical quasi-Gaussian beams.

Making use of the complex-source-point method in cylindrical coordinates, an exact solution representing a cylindrical quasi-Gaussian beam of arbitrar...
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