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On the beam shape coefficients of fundamental nondiffracting beams
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On the beam shape coefficients of fundamental nondiffracting beams A. Chafiq , G. Gouesbet , A. Belafhal PII: DOI: Reference:
S0022-4073(19)30724-1 https://doi.org/10.1016/j.jqsrt.2019.106750 JQSRT 106750
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
6 October 2019 7 November 2019 7 November 2019
Please cite this article as: A. Chafiq , G. Gouesbet , A. Belafhal , On the beam shape coefficients of fundamental nondiffracting beams, Journal of Quantitative Spectroscopy & Radiative Transfer (2019), doi: https://doi.org/10.1016/j.jqsrt.2019.106750
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HIGHLIGHTS
A new method of evaluating the beam shape coefficients (BSCs) in generalized Lorenz-Mie theory for fundamental nondiffracting beams, cosine beams, Bessel beams, Mathieu beams and parabolic beams is investigated. The relationship between solutions of scalar Helmholtz equation to expand the BSCs of nondiffracting beams in terms of higher order Bessel Beams BSCs is exploited. For validation of the results, numerical simulations and discussions are given.
1
On the beam shape coefficients of fundamental nondiffracting beams A. Chafiq1,2, G. Gouesbet3, A. Belafhal1* 1
3
Laboratory LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B 20, 24000 El Jadida, Morocco 2 CRMEF Marrakech-Safi annexe de Safi, 46000 Safi, Morocco
UMR 6614/CORIA, CNRS, Université et INSA de Rouen, Site du Madrillet, Avenue de l’Université, BP12 76801 Saint Etienne du Rouvray, France * Corresponding authors: [email protected]
Abstract In generalized Lorenz-Mie theory (GLMT) applied to structured beams the evaluation of beam shape coefficients (BSCs) constitutes a challenge. In this paper, we propose to calculate BSCs of fundamental nondiffracting beams, cosine beams, Bessel beams, Mathieu beams and parabolic beams. The aim of our method is the use of the Whittaker integral related to a scalar nondiffracting beam, and well known angular spectrum decomposition. Also, we exploit the relationship between solutions of scalar Helmholtz equation to expand the BSCs of nondiffracting beams in terms of higher order Bessel Beams (HOBB) BSCs. Some numerical simulations and discussions are given to validate our results.
Keywords: Generalized Lorenz-Mie theory; Beam shape coefficients; Nondiffracting beams; Cosine beams; Higher order Bessel beams; Mathieu beams; Parabolic beams; Helmholtz equation.
1. Introduction Among theories that describe interaction between light and particles, we consider Generalized Lorenz-Mie Theory (GLMT) [1-3] and Extended Boundary Condition Method (EBCM) [4]. In these theories beams are encoded by coefficients that depend on the shape of the beam, usually denoted as g nm, TM and g nm, TE and called transverse magnetic (TM) and transverse electric (TE) Beam Shape Coefficients (BSCs). These coefficients serve to expand beams in terms of Vector Spherical Wave Functions (VSWFs). Originally, BSCs are evaluated by using quadrature methods [1], a finite series technique [5], the angular spectrum decomposition (ASD) method [6] and localized approximations [7-8]. Unfortunately, the quadrature methods are time-consuming due to the fact that the kernel to integrate is highly oscillating. Conversely, the use of localized approximations revealed to be the most efficient one in the case of Gaussian beams [7,9], recently reviewed in [8] and [10]. In Ref. [11, 12], 2
Gouesbet and Lock have distinguished between two methods in evaluating BSCs, namely intrinsic and extrinsic methods. This classification is made according to whether methods to evaluate BSCs of GLMT posed in a certain coordinate system are carried out in terms of quantities pertaining to the same coordinate system or to a different coordinate system. Recently, an intrinsic method has been extended to the oblique illumination in spherical coordinates in Ref. [13]. Originally, BSCs were expressed by using the spherical harmonic expansion technique. However, in ASD method, the beam is expressed with spectral components which can be obtained from Fourier transforms of EM fields. In Ref. [11], the relationship between the BSCs and the plane wave spectra has been established. Since the development of GLMT, strong effort has been devoted to calculate BSCs of arbitrary shaped beams. Beside evaluating conventional BSCs of plane waves and of Gaussian beams, many BSCs have been calculated for other kinds of beams as zeroth-order Bessel beams [14] and their continuous superpositions [15], Higher Order Bessel Beams [16] and their discrete superpositions [17-19], zeroth-order Mathieu beams [20], and recently Laguerre-Gauss beams [21-23]. For nondiffracting beams it has been demonstrated that localized approximations, including integral localized approximations, provide a satisfactory description of the intended beam only if the axicon angle is small enough and/or a net topological charge not exists [20, 24-26]. Parallel efforts have been devoted in designing optical beams to investigate news aspect of interaction between light and particles. In 1987, the term of nondiffracting beam appeared in a work of Durnin [27-28] on the propagation in vacuum of special beam called Bessel beams. In this original paper, the beams were described as exact solutions to the homogeneous Helmholtz equation. Later, more general types of nondiffracting beams were introduced as cosine, HOBB, Mathieu, Lommel and parabolic beams [29-34]. Since the first works of Durnin, there has been much research on nondiffracting beams and their apertured or Gaussian modulated versions concerning their generation [35-39], propagation [32, 40-42], applications [43-49] etc. Nondiffracting optical beams are known to propagate indefinitely without change of their transverse shape making them different from the conventional laser beams profiles. A common theoretical way to obtain nondiffracting beams is by solving propagation equation by the method of separation of variable. In this approach, the propagation variable is involved in an exponential term and the other transverse coordinates are expressed in a function that obey the two dimensional Helmholtz equation (HE). In [50], Miller gives a method for exploiting the relationship between separable solutions of HE. According to this result, the transverse profile of any nondiffracting beam can be written in 3
term of Whittaker integral in cylindrical coordinates by using the angular spectrum decomposition [50], which is derived by applying the Fourier transform on solutions of HE. This idea is exploited in [32] and [51] to expand Mathieu beams and parabolic beams (or Weber beams) in the basis of Bessel solutions. In this work, we will exploit the relationship between separable solutions of HE to derive a relationship between BSCs of nondiffracting beam and BSCs of Bessel Beam, the later being known from some works as [14, 16, 52, 53]. The paper is organized as follows: in Section 2, we recall the theoretical background concerning exact solutions of HE and angular spectrum related to these solutions, and the expansion of solutions in terms of Bessel functions basis is given. Section 3 presents the theory for deriving relationship between BSCs of nondiffracting beams and BSCs of HOBBs. Section 4 is devoted to a conclusion.
2. Background In this section, we start with a standard derivation of solutions of the scalar electromagnetic wave equation, assuming a monochromatic (time-harmonic) electromagnetic wave of frequency ω, with time dependence of the form expit . Maxwell’s equations for the scalar electrical field E r , in the absence of free currents and free charges, then obey the tridimensional Helmholtz equation E r k 2 E r 0 ,
(1)
where is the tridimensional Laplacian operator and k is the wavenumber of wave vector. The method of separation of variables for solving this differential equation can be applied. Indeed, it is well known that this equation is separable in eleven coordinates systems [50,54]. For the cylindrical and Cartesian coordinate systems, the solution can be written as
E (r , z) U (r ) expik z z ,
(2)
where k z is the longitudinal wavenumber related to the transverse wavenumber kt by
k z k 2 k t2 , r x , y , denotes the transverse coordinates, x, y are Cartesian coordinates and , are cylindrical coordinates. U r is solution of the two dimensional Helmholtz equation
U r k t2U (r ) 0 ,
(3)
where is the transverse Laplacian operator. Solutions of Eq. (3) can be written involving Fourier transform as 4
U x, y
U k
x
, k y exp i k x x k y y dk x dk y ,
(4.a)
where U k x , k y is Fourier transform of U x, y . Transverse distribution of beams U x, y verify two dimensional HE if k t2 k x2 k y2 and
U k x , k y
1 k t k A , kt
(4.b)
where A is known as the angular spectrum of beams, k t k is Dirac distribution, k
and are modulus of transverse wave vector and angular variable in k x , k y
plane,
respectively. After integration over transverse wave vector, the transverse distribution of nondiffracting beams in Cartesian coordinate system becomes
U x, y
A expik x cos y sin d . t
(4.c)
This result can be rewritten in cylindrical coordinate system as
U ,
A expik cos d .
(4.d)
t
Integrals in Eqs. (4-c,d) are known as Whittaker integrals. A geometrical interpretation of these integrals implies that any nondiffracting beam is a coherent superposition of plane waves whose propagation vectors cover the conical surface with the vertex angle
arcsin
kt , thus being called the axicon angle [55,56]. The amplitudes and relative kz
phases of the superposed plane waves can be arbitrary. They are described by the function
A . Due to that arbitrariness, an infinite number of nondiffracting beams with different transverse intensity profiles can be obtained. In this geometrical interpretation, k t and k z represent projections of the propagation vectors of the plane wave components of the wave vector onto the transverse plane (x, y) and to the direction of propagation coinciding with the z axis, respectively. They can be expressed as k t k sin and k z k cos . From expressions of A , we can distinguish some kinds of nondiffracting beams defining conventional (or fundamental) nondiffracting beams for which the angular spectrum A is given by a well-
5
known mathematical functions. This category of beams includes: plane waves (or cosine beams), Bessel beams, Mathieu beams and parabolic beams. In the case where A is an arbitrary function we can define other kinds of beams called random nondiffracting beams [57]. In Ref. [50], Miller considered the conventional nondiffracting beams as the separable solutions of Helmholtz equation and established relationships between them. The expansion of these solutions in plane wave basis gives the Whittaker integrals recalled in Eqs. (4-c,d). In this work, we use the same expansion of nondiffracting beams but in the discrete Bessel solutions basis i.e
U NB r
C
NB l
U lBB r ,
(5.a)
l
with, using both bra-ket and integral notations
ClNB A NB , A BB A NB A
BB
d .
(5.b)
ANB A
BB
is
the
angular
spectrum
of
a
given
nondiffracting
beam
and
1 exp il is the angular spectrum complex conjugate of BBs. U NB r is 2
the transverse field distribution of nondiffracting beams and U lBB r is the field distribution of HOBBs given by [50] U lBB r i l 2 J l k t exp(il ) .
(6)
The expansion of Eq. (5.a) is used in various Refs. [32, 34, 51, 58] to express nondiffracting beams in terms of BBs. Furthermore, vectorial solutions of HE can be built from scalar nondiffracting fields by using some approaches as in Refs. [59] and [60]. Following the first approach of Ref. [59], a transverse vector HE admits two kinds of solutions
E t1 (rt ) U rt ,
(7.a)
and
E t2 (rt ) e z E t1 (rt ) ,
(7.b)
and the longitudinal components of vector field are given by
1 Ez1,2 (rt ) . E t1, 2 (rt ) . k
(7.c)
6
In these expressions U rt is the scalar electric field of nondiffracting beams. General vector solutions can be expressed as linear combination of E 1 (r) and E 2 (r ) . The magnetic vector field can be deduced from the electric field vector and from Maxwell equations. Eqs. (7) give us an easy way to deduce vector fields from scalar fields. Since Eqs. (7) use only linear operators, then if the scalar beam field is expanded in scalar HE eigenfunctions basis, the obtained vector beam field is also an expansion of vector HE basis. This conclusion allows us to write the following expressions for electric and magnetic vector fields of nondiffracting beams as
E ND ( , , z )
C
ND l
E lBB ( , , z ) ,
(8.a)
ND l
H lBB ( , , z ) ,
(8.b)
l
and
H ND ( , , z )
C l
where E lBB and H lBB are the electric and magnetic vector fields of HOBBs. In the following these expressions will be used to derive the beam shape coefficients.
3. Beam shape coefficients of conventional nondiffracting beams In the framework of GLMT and in spherical coordinates, the electric and magnetic fields E and H of an arbitrary shaped beam, with a time-dependence of the form expit , can be expanded in terms of VSWFs [1-4]. In particular, the radial components of the electric and magnetic fields play an important role in this framework because they allow one to determine the BSCs. These radial components are given by
Er E0
n
C
pw n
g nm,TM
n 1 m n
nn 1 m j n kr Pn cos expim , r
(9.a)
nn 1 m j n kr Pn cos exp im , r
(9.b)
and
Hr H0
n
C
n 1 m n
pw n
g nm,TE
7
where jn kr is the first kind spherical Bessel function, k is the wavenumber and Pnm cos is the associated Legendre function given by [61]
Pn
m
m cos 1m sin m d Pn cosm , d cos
(9.c)
and Cnpw (with pw standing for “plane wave”) denotes expansion coefficients which appear in the Bromwich formulation of the classical Lorenz Mie theory [2] according to
C
pw n
n 1 i
k
2n 1 . nn 1
(9.d)
In Eq. (9.a-b), BSCs may be expressed in terms of radial electric and magnetic fields components by using quadrature methods [62] i.e g nm,TM i n 1 n m ! kr 2 E r kr / E 0 m im sin dd . m H kr / H Pn cos e 0 g n ,TE 4 n m ! j n kr 0 0 r
(10)
By inserting Eqs.(8.a-b) into Eq.(10), BSCs of nondiffracting beams can easily be deduced as
ND m n ,TM ND m
g g n ,TE
ND m C l g n,TM l C ND g m l n ,TE l
where g nm,TM
BB
l
and g nm,TE
BB
l
BB
l
BB
l
,
(11)
are BSCs of HOBBs.
The use of this result depends on our ability to calculate the coefficients ClND and the BSCs
g
BB m n ,TM l
and g nm,TE
BB
l
. Calculations of the first kind ClND of coefficients for fundamental
nondiffracting beams will be given in the sequel. For the second kind of coefficients, namely BSCs of BBs, Mitri in [63] calculated them numerically by using double quadratures. For analytical calculations, Taylor and Love [52] used the angular spectrum decomposition to derive BSCs for zeroth-order Bessel beams. The same procedure was used by Lock [14] to analyze BSCs of zeroth-order Bessel beams and by Chen et al [53] to determine BSCs of polarized BBs of arbitrary order. Recently, Wang et al [64] derived a general description of higher order Bessel beams, which take into account results concerning Davis Bessel beams
8
type. According to this work and [16], TE and TM BSCs for circularly symmetric BB are given by gm n , TM TE
BB
m m 1 g 1m m / 2 n m ! e ik z z0 i l m 1 e i l m 1 0 J l m 1 k t 0 n cos m n cos i n m ! l 1 i l m 1e i l m 1 0 J l m 1 k t 0 nm cos m nm cos , 1
(12.a)
where nm and nm are the generalized Legendre functions given by
nm cos
Pnm cos , sin
(12.b)
dPnm cos . d
(12.c)
and
nm cos
In Eq. (12.a), g 1 cos / 4 is introduced to reduce BBs to Davis Bessel beams used in [16, 65]. 0 , 0 , z0 are the cylindrical coordinates of the beam center in the used coordinate system. is the axicon angle mentioned above in geometrical interpretation of Whittaker integrals. In Fig.1, we reproduce plots of Fig. (2.a-b) of [16] of BSCs of second order BBs for m=1,3. It should be noted that for an on axis BBs of l-order all BSCs are zero except for
m l 1 , corresponding to m 1 and m 3 for l 2 . Thus, in the series of Eq. (11) and for a given value of m only terms corresponding to l m 1 are different from zero. BSCs of any
nondiffracting beam are then obtained by multiplying g nm,TM
m l 1 , we have g nm,TM
BB
m 1
i g nm,TE
BB
m 1
BB
m 1
by CmND1 . Furthermore, for
. In the sequel, this example of BBs will be used to
calculate BSCs for other fundamental nondiffracting beams (plane wave, cosine, Mathieu and parabolic beams). But the application of Eq. (11) for these beams depends on the calculation of overlap coefficients ClND and on the convergence of series in Eq. (5.a), Eqs. (8.a-b) and Eq. (11) (for more discussion on convergence of BSCs of BBs see [16] and [32, 51] for the convergence of the series in Eq. (11)).
9
(a)
Figure 1: g nm,TM
(b)
BB
l
of BBs for an axicon angle 10 , 20° and 30°:
(a) l 2, m 1 ; (b) l 2, m 3 .
3.1 Plane wave and cosine beams Homogeneous plane waves are solutions of HE in Cartesian coordinates. The electric field and the angular spectrum associated with these beams are, respectively
U PW x, y U 0 expik x x expik y y ,
(13.a)
and
APW ,
(13.b)
where is a constant angle. In this case k x kt cos and k y k t sin . Some patterns can be
obtained
from
superposition
of
plane
waves.
For
example,
if
we
take
ACB , we obtain cosine beams, introduced by Jiang in Ref. [29]. 2 2 The electric field of these beams is
U CB x, y U 0 exp ik y y exp ik y y 2U 0 cosk y y .
10
(13.c)
Two-dimensional intensity distribution of these beams is given in Fig. 2. The overlap coefficients between plane wave or cosine beam and Bessel beams can be obtained by inserting the angular spectrum of these beams in Eq. (5.b). ClPW and ClCB are then given by
ClPW
1 exp il , 2
(13.d)
2 cos l . 2
(13.e)
and
ClCB
Since C lCB C CBl , Eq. (5.a) can be written as
U CB r 2 0,l ClCB (i ) l 2 J l k t cos(l ) .
(13.f)
l 0
Fig. (2.b) displays a comparison between the formulation of scalar Cosine beam using Eq.(13.c) and the expansion formulation in terms of Bessel beams basis by using the coefficient ClCB (Eq.(13.f)), From this figure, we observe a good agreement between the two
formulations of cosine beams. In Fig (2.c), we plot g1n,TM
CB
versus n, where g1n,TM
CB
are
calculated by expanding Cosine beams in terms of circularly symmetric BBs such as described in [16].
11
(a)
(b)
(c) Figure 2: Cosine beams: (a) Two dimensional intensity distribution of a cosine beam with k y L 5 (the size of transverse display area is L L ); (b) Comparison between intensity of a cosine beam using formulation of Eq. (13.c) and BB expansion of Eq. (13.f); (c) BSCs performed with Eq. (11) for axicon angles 10 , 20° and 30°.
3.2 Higher order Bessel Beams For a m-th order BBs, the angular spectrum is [50, 66]:
A BB
1 2
expim .
(14.a)
In this case the overlap coefficients are given by
12
ClBB
1 2
2
expim exp il d 0
lm
0 if m l , 1 if m l
(14.b)
where lm is the Kronecker index. BSCs of these beams are well known from various works as [14,16,52,53,63].
3.3 Mathieu beams In elliptic cylindrical coordinates, the transverse HE can be split in two differential equations: the radial Mathieu equation and the angular Mathieu equation. Solutions of these equations are radial and angular Mathieu functions, respectively. In terms of these functions, the even and odd fields distributions are given by
U e r U 0e Jem , q cem , q ; m=0,1,2 … for even modes,
(15.a)
and U o r U 0 o Jo m , q se m , q ; m=1,2, … for odd modes,
(15.b)
where Jem and Jom are mth-order even and odd radial Mathieu functions, respectively. cem and so m are mth-order even and odd angular Mathieu functions, respectively.
q
h 2 k t2 is the ellipticity parameter and h is the semi-confocal parameter. , denote 2
transverse
elliptic
coordinates
related
to
the
transverse
coordinates
x, y
by
x h cosh cos and y h sinh sin . The angular spectra of Mathieu beams are the
angular Mathieu function cem ( , q) for even modes and sem ( , q) for odd modes [31, 32, 50, 66], i.e MB Ae AoMB
1
1
cem ( , q ) sem ( , q )
1
1
A q cos j (m) j
j 0
.
(15.c)
B q sin j (m) j
j 1
A(jm ) q and B (jm ) q are the expansion coefficients of Mathieu functions. By inserting
AeMB and AOMB in Eq. (5.b), the coefficients ClND corresponding to Mathieu beams modes are given by 13
CeMB ,l
A(j m) if l j 0 2 A( m) j if l j 0 2 2 otherwise 0
CoMB ,l
B (j m) if l j i 2 0 otherwise
for even modes,
(15.d)
for odd modes.
(15.e)
By using these coefficients one can expand the field and the BSCs of Mathieu beams in terms of BBs and BSCs of these later beams, respectively. Using these ClMD , Mathieu beams fields can be written as:
U eMB r 2 A (j m ) (i ) l J l k t cosl for even modes,
(15.f)
j 0
and
U oMB r 2 B (j m) (i) l J l k t sin l for even modes.
(15.g)
j 1
The above results was already used in Ref. [32] for studying paraxial propagation of Mathieu beams and in Ref. [20], for evaluating the validity of localized integral
approximation in the case of zeroth-order Mathieu beams. In Fig. (3), we present g nm,TM
MB
versus n from Eq. (11), for zeroth order Mathieu beams and fourth order even Mathieu beams. From these figures, it can be seen that the BSCs are very well defined for n ranging from 0 to 150.
14
(a)
(b)
Figure 3: Evolution of BSCs versus n of Mathieu beams for q=20, m=1, axicon angles
10 , 20° and 30°: (a) zeroth-order Mathieu beams; (b) Fourth order even Mathieu beams.
3.4 Parabolic beams or Weber beams In literature, there are various representations of transverse separable solutions of HE expressed in cylindrical parabolic coordinates. In [33], Bandres et al. presented the solution as a product of parabolic cylinder functions Pe and Po , which are two series of parabolic cylinder coordinates. In Refs. [50, 51, 54, 61, 67] transverse solutions are connected with Weber differential equation whose solutions are expressed with parabolic cylinder functions D . In the work of Rodríguez-Lara [51], the expression of parabolic beams is given by 2
U ePB a , r U 0e
1 ia 4 2 U e a, u Ve a, v 2 sin
even mod es ,
(16.a)
odd mod es ,
(16.b)
and 2
U oPB a, r U 0o
3 ia 2 4 2 U O a, u VO a , v sin
where u, v are parabolic coordinates related to Cartesian coordinates by u iv 2 x iy , a 1/ 2
is a separation constant. U e ,o a, u and Ve,o a, u are given by
15
1 ia 1 U e a, u exp ik t u 2 / 2 1 F1 , , ik t u 2 , 4 2 2
(16.c)
1 ia 1 Ve a, v exp ik t v 2 / 2 1F1 , , ik t v 2 , 4 2 2
(16.d)
3 ia 3 U e a, u 2k t u exp ik t u 2 / 2 1 F1 , , ik t u 2 , 4 2 2
(16.e)
and 3 ia 3 Ve a, v 2k t v exp ik t v 2 / 2 1F1 , , ik t v 2 , 4 2 2
(16.f)
where 1 F1 is a Hypergeometric function. In [50, 66, 67], the angular spectra arising from the plane wave decomposition of even and odd modes of parabolic beams are
exp ia ln(tan ) 2 , AePB a, 2 sin
(16.g)
and
AoPB a, i sgn AePB a, .
(16.h)
Even and odd solutions are constructed from the sum or the difference of fields obtained from the Whittaker integral. By inserting the above angular spectra in Eq. (5.b), the overlap coefficients corresponding to parabolic beams can be written as
C ePB ,l
1 2
a 1 i 2 4
1 cos
a 1
1 cos i 2 4 exp(il )d ,
(17.a)
and
CoPB,l
a 1 a 1 i i i 1 cos 2 4 1 cos 2 4 exp(il )d 2 0 0
a 1 i 2 4
1 cos
a 1 i 2 4
1 cos
exp(il )d
16
.
(17.b)
By using results from [51], and result 3.892. 3 p .486 of [61], overlap coefficients are given by (here we use the same notation as in [51]) C ePB ,l
1 2 a i ) / 4 C l , a C l , a , e 2
C oPB,l
(18.a)
i 2 a i ) / 4 e C l , a C l , a , 2
(18.b)
with 1 l C l , a l 1 f l , a i sgn a f l ,a , 2
(18.c)
and
1 2 ia ia 2 F 1 ia , 1 ia;1 l ia ; 1 , f l , a 2 1 1 l ia 2 2 2
(18.d)
where 2 F1 denotes a Hypergometric function. In [51], Rodríguez-Lara demonstrated that the contribution of overlap coefficients increases as l increases until a given l and afterward it oscillates around a maximum value of these coefficients. This structural property permits to conclude that it is not feasible to efficiently construct a parabolic beam through a finite superposition of just a few Bessel beams. Recently, Khonina et al [68-69] expanded the angular spectra function of the even modes with an additional index p (integer or fractional) in Fourier series as 1 2 2 AePB a, a le a 1 0 ,s cos s bse a cos s , 2 T T s 0
(19.a)
where as , bs are Fourier coefficients defined by T/2
a se a
2 2 AePB d , , o a, cos s T T / 2 T
(19.b)
and T /2
bse a
2 2 AePB d . , o a , sin s T T / 2 T
(19.c)
The period of the function AePB , o a, is T 2 . Relying on the use of the additional index p, Khonina et al. distinguished between the case a 0 and the case a 0 to develop this beam 17
in terms of Bessel beams. For p 1 and for a 0 , the authors demonstrated in [68] that bse 0 0 and
a se 0
1s
2 . 3 3 s s 4 4
(19.d)
Substituting the Fourier decomposition in Eq. (4.c), we deduce
U ePB a, r U 0e exp ikz 2 0, s i a se 0 coss J s k t . s
(19.e)
s0
For the case a 0 , the obtained integral expressions for a se a and bse a have to be calculated numerically, insofar as it is difficult to reduce them to any closed formula. Furthermore, even if we calculate a se and bse the Fourier series cannot be used in this case because of their slow convergence as for Lara approach [51]. In the following, we therefore contented ourselves to only study the case a 0 . Fig. (3) confirms that, in the case a 0 , the series of Eq. (19.e) converge and BSCs can be calculated from those of BB. By inserting AePB a 0, into Eq. (5.b), the overlap coefficients are given by
C
PB l
e 1 a s (0)1 0, s coss exp il d 2 s 0 2 1
.
e 1 a 0 1 if l s 2 l 2 0,l 0 otherwise
(20)
With these overlap coefficients we easily deduce the expansion series in Eq. (19.e) of parabolic beams in terms of BBs expressed from Eq. (6). For parabolic beams odd modes, we use AoPB a, of Eq. (16.g) to derive similar overlap coefficients. In Fig. (4), we plot
parabolic beams for a 0 and its g nm,TM
PB
versus n calculated from Eq. (11).
18
(a)
(b)
(c) Figure 4: Parabolic beam when a 0 : (a) Intensity of beam computed by the exact solution (Eq.4.c); (b) Intensity of beam computed by the finite series solution (Eq.(19.e); (c) Evolution of BSCs versus n for m=1 and for axicon angles 10 , 20° and 30°.
From these Figures, we deduce that the Bessel expansion gives the same results as the numerical calculation of the intensity from Whittaker integrals. Curves in Fig. (4.c) are validation of Bessel expansion for BSCs. 4. Conclusion In this work, we have introduced a new method of calculating the BSCs used in GLMT for fundamental scalar or vector nondiffracting beams. We have exploited the relationship between separable exacts solutions of Helmholtz equation. Theoretical study shows that any exact solution can be expressed as a sum or integral over basis functions formed by other 19
exact solutions. Whittaker integrals express any exact solutions of Helmholtz equation in terms of plane waves. In this paper, we have expressed exact solutions of Helmholtz equation i.e nondiffracting beams as sum of BBs of various orders. The expansion coefficients are named overlap coefficients which are calculated from angular spectra of nondiffracting beams and BBs. The overlap coefficients are used to expand BSCs of nondiffracting beams in terms of BSCs of BBs. The proposed method can be used for all conventional nondiffracting beams except for parabolic beams for a 0 , because of the slow convergence of the expansion series. As application of this method, we have used BSCs of circularly symmetric BB to derive BSCs of plane wave, cosine, Mathieu and parabolic beams ( a 0 ) and plot the evolution of these BSCs versus n. Finally, the proposed method can enrich the possibility of employing the GLMT in studying various optical applications that use nondiffracting beams.
20
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On the beam shape coefficients of fundamental nondiffracting beams A. Chafiq , G. Gouesbet , A. Belafhal PII: DOI: Reference:
S0022-4073(19)30724-1 https://doi.org/10.1016/j.jqsrt.2019.106750 JQSRT 106750
To appear in:
Journal of Quantitative Spectroscopy & Radiative Transfer
Received date: Revised date: Accepted date:
6 October 2019 7 November 2019 7 November 2019
Please cite this article as: A. Chafiq , G. Gouesbet , A. Belafhal , On the beam shape coefficients of fundamental nondiffracting beams, Journal of Quantitative Spectroscopy & Radiative Transfer (2019), doi: https://doi.org/10.1016/j.jqsrt.2019.106750
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HIGHLIGHTS
A new method of evaluating the beam shape coefficients (BSCs) in generalized Lorenz-Mie theory for fundamental nondiffracting beams, cosine beams, Bessel beams, Mathieu beams and parabolic beams is investigated. The relationship between solutions of scalar Helmholtz equation to expand the BSCs of nondiffracting beams in terms of higher order Bessel Beams BSCs is exploited. For validation of the results, numerical simulations and discussions are given.
1
On the beam shape coefficients of fundamental nondiffracting beams A. Chafiq1,2, G. Gouesbet3, A. Belafhal1* 1
3
Laboratory LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Chouaïb Doukkali University, P. B 20, 24000 El Jadida, Morocco 2 CRMEF Marrakech-Safi annexe de Safi, 46000 Safi, Morocco
UMR 6614/CORIA, CNRS, Université et INSA de Rouen, Site du Madrillet, Avenue de l’Université, BP12 76801 Saint Etienne du Rouvray, France * Corresponding authors: [email protected]
Abstract In generalized Lorenz-Mie theory (GLMT) applied to structured beams the evaluation of beam shape coefficients (BSCs) constitutes a challenge. In this paper, we propose to calculate BSCs of fundamental nondiffracting beams, cosine beams, Bessel beams, Mathieu beams and parabolic beams. The aim of our method is the use of the Whittaker integral related to a scalar nondiffracting beam, and well known angular spectrum decomposition. Also, we exploit the relationship between solutions of scalar Helmholtz equation to expand the BSCs of nondiffracting beams in terms of higher order Bessel Beams (HOBB) BSCs. Some numerical simulations and discussions are given to validate our results.
Keywords: Generalized Lorenz-Mie theory; Beam shape coefficients; Nondiffracting beams; Cosine beams; Higher order Bessel beams; Mathieu beams; Parabolic beams; Helmholtz equation.
1. Introduction Among theories that describe interaction between light and particles, we consider Generalized Lorenz-Mie Theory (GLMT) [1-3] and Extended Boundary Condition Method (EBCM) [4]. In these theories beams are encoded by coefficients that depend on the shape of the beam, usually denoted as g nm, TM and g nm, TE and called transverse magnetic (TM) and transverse electric (TE) Beam Shape Coefficients (BSCs). These coefficients serve to expand beams in terms of Vector Spherical Wave Functions (VSWFs). Originally, BSCs are evaluated by using quadrature methods [1], a finite series technique [5], the angular spectrum decomposition (ASD) method [6] and localized approximations [7-8]. Unfortunately, the quadrature methods are time-consuming due to the fact that the kernel to integrate is highly oscillating. Conversely, the use of localized approximations revealed to be the most efficient one in the case of Gaussian beams [7,9], recently reviewed in [8] and [10]. In Ref. [11, 12], 2
Gouesbet and Lock have distinguished between two methods in evaluating BSCs, namely intrinsic and extrinsic methods. This classification is made according to whether methods to evaluate BSCs of GLMT posed in a certain coordinate system are carried out in terms of quantities pertaining to the same coordinate system or to a different coordinate system. Recently, an intrinsic method has been extended to the oblique illumination in spherical coordinates in Ref. [13]. Originally, BSCs were expressed by using the spherical harmonic expansion technique. However, in ASD method, the beam is expressed with spectral components which can be obtained from Fourier transforms of EM fields. In Ref. [11], the relationship between the BSCs and the plane wave spectra has been established. Since the development of GLMT, strong effort has been devoted to calculate BSCs of arbitrary shaped beams. Beside evaluating conventional BSCs of plane waves and of Gaussian beams, many BSCs have been calculated for other kinds of beams as zeroth-order Bessel beams [14] and their continuous superpositions [15], Higher Order Bessel Beams [16] and their discrete superpositions [17-19], zeroth-order Mathieu beams [20], and recently Laguerre-Gauss beams [21-23]. For nondiffracting beams it has been demonstrated that localized approximations, including integral localized approximations, provide a satisfactory description of the intended beam only if the axicon angle is small enough and/or a net topological charge not exists [20, 24-26]. Parallel efforts have been devoted in designing optical beams to investigate news aspect of interaction between light and particles. In 1987, the term of nondiffracting beam appeared in a work of Durnin [27-28] on the propagation in vacuum of special beam called Bessel beams. In this original paper, the beams were described as exact solutions to the homogeneous Helmholtz equation. Later, more general types of nondiffracting beams were introduced as cosine, HOBB, Mathieu, Lommel and parabolic beams [29-34]. Since the first works of Durnin, there has been much research on nondiffracting beams and their apertured or Gaussian modulated versions concerning their generation [35-39], propagation [32, 40-42], applications [43-49] etc. Nondiffracting optical beams are known to propagate indefinitely without change of their transverse shape making them different from the conventional laser beams profiles. A common theoretical way to obtain nondiffracting beams is by solving propagation equation by the method of separation of variable. In this approach, the propagation variable is involved in an exponential term and the other transverse coordinates are expressed in a function that obey the two dimensional Helmholtz equation (HE). In [50], Miller gives a method for exploiting the relationship between separable solutions of HE. According to this result, the transverse profile of any nondiffracting beam can be written in 3
term of Whittaker integral in cylindrical coordinates by using the angular spectrum decomposition [50], which is derived by applying the Fourier transform on solutions of HE. This idea is exploited in [32] and [51] to expand Mathieu beams and parabolic beams (or Weber beams) in the basis of Bessel solutions. In this work, we will exploit the relationship between separable solutions of HE to derive a relationship between BSCs of nondiffracting beam and BSCs of Bessel Beam, the later being known from some works as [14, 16, 52, 53]. The paper is organized as follows: in Section 2, we recall the theoretical background concerning exact solutions of HE and angular spectrum related to these solutions, and the expansion of solutions in terms of Bessel functions basis is given. Section 3 presents the theory for deriving relationship between BSCs of nondiffracting beams and BSCs of HOBBs. Section 4 is devoted to a conclusion.
2. Background In this section, we start with a standard derivation of solutions of the scalar electromagnetic wave equation, assuming a monochromatic (time-harmonic) electromagnetic wave of frequency ω, with time dependence of the form expit . Maxwell’s equations for the scalar electrical field E r , in the absence of free currents and free charges, then obey the tridimensional Helmholtz equation E r k 2 E r 0 ,
(1)
where is the tridimensional Laplacian operator and k is the wavenumber of wave vector. The method of separation of variables for solving this differential equation can be applied. Indeed, it is well known that this equation is separable in eleven coordinates systems [50,54]. For the cylindrical and Cartesian coordinate systems, the solution can be written as
E (r , z) U (r ) expik z z ,
(2)
where k z is the longitudinal wavenumber related to the transverse wavenumber kt by
k z k 2 k t2 , r x , y , denotes the transverse coordinates, x, y are Cartesian coordinates and , are cylindrical coordinates. U r is solution of the two dimensional Helmholtz equation
U r k t2U (r ) 0 ,
(3)
where is the transverse Laplacian operator. Solutions of Eq. (3) can be written involving Fourier transform as 4
U x, y
U k
x
, k y exp i k x x k y y dk x dk y ,
(4.a)
where U k x , k y is Fourier transform of U x, y . Transverse distribution of beams U x, y verify two dimensional HE if k t2 k x2 k y2 and
U k x , k y
1 k t k A , kt
(4.b)
where A is known as the angular spectrum of beams, k t k is Dirac distribution, k
and are modulus of transverse wave vector and angular variable in k x , k y
plane,
respectively. After integration over transverse wave vector, the transverse distribution of nondiffracting beams in Cartesian coordinate system becomes
U x, y
A expik x cos y sin d . t
(4.c)
This result can be rewritten in cylindrical coordinate system as
U ,
A expik cos d .
(4.d)
t
Integrals in Eqs. (4-c,d) are known as Whittaker integrals. A geometrical interpretation of these integrals implies that any nondiffracting beam is a coherent superposition of plane waves whose propagation vectors cover the conical surface with the vertex angle
arcsin
kt , thus being called the axicon angle [55,56]. The amplitudes and relative kz
phases of the superposed plane waves can be arbitrary. They are described by the function
A . Due to that arbitrariness, an infinite number of nondiffracting beams with different transverse intensity profiles can be obtained. In this geometrical interpretation, k t and k z represent projections of the propagation vectors of the plane wave components of the wave vector onto the transverse plane (x, y) and to the direction of propagation coinciding with the z axis, respectively. They can be expressed as k t k sin and k z k cos . From expressions of A , we can distinguish some kinds of nondiffracting beams defining conventional (or fundamental) nondiffracting beams for which the angular spectrum A is given by a well-
5
known mathematical functions. This category of beams includes: plane waves (or cosine beams), Bessel beams, Mathieu beams and parabolic beams. In the case where A is an arbitrary function we can define other kinds of beams called random nondiffracting beams [57]. In Ref. [50], Miller considered the conventional nondiffracting beams as the separable solutions of Helmholtz equation and established relationships between them. The expansion of these solutions in plane wave basis gives the Whittaker integrals recalled in Eqs. (4-c,d). In this work, we use the same expansion of nondiffracting beams but in the discrete Bessel solutions basis i.e
U NB r
C
NB l
U lBB r ,
(5.a)
l
with, using both bra-ket and integral notations
ClNB A NB , A BB A NB A
BB
d .
(5.b)
ANB A
BB
is
the
angular
spectrum
of
a
given
nondiffracting
beam
and
1 exp il is the angular spectrum complex conjugate of BBs. U NB r is 2
the transverse field distribution of nondiffracting beams and U lBB r is the field distribution of HOBBs given by [50] U lBB r i l 2 J l k t exp(il ) .
(6)
The expansion of Eq. (5.a) is used in various Refs. [32, 34, 51, 58] to express nondiffracting beams in terms of BBs. Furthermore, vectorial solutions of HE can be built from scalar nondiffracting fields by using some approaches as in Refs. [59] and [60]. Following the first approach of Ref. [59], a transverse vector HE admits two kinds of solutions
E t1 (rt ) U rt ,
(7.a)
and
E t2 (rt ) e z E t1 (rt ) ,
(7.b)
and the longitudinal components of vector field are given by
1 Ez1,2 (rt ) . E t1, 2 (rt ) . k
(7.c)
6
In these expressions U rt is the scalar electric field of nondiffracting beams. General vector solutions can be expressed as linear combination of E 1 (r) and E 2 (r ) . The magnetic vector field can be deduced from the electric field vector and from Maxwell equations. Eqs. (7) give us an easy way to deduce vector fields from scalar fields. Since Eqs. (7) use only linear operators, then if the scalar beam field is expanded in scalar HE eigenfunctions basis, the obtained vector beam field is also an expansion of vector HE basis. This conclusion allows us to write the following expressions for electric and magnetic vector fields of nondiffracting beams as
E ND ( , , z )
C
ND l
E lBB ( , , z ) ,
(8.a)
ND l
H lBB ( , , z ) ,
(8.b)
l
and
H ND ( , , z )
C l
where E lBB and H lBB are the electric and magnetic vector fields of HOBBs. In the following these expressions will be used to derive the beam shape coefficients.
3. Beam shape coefficients of conventional nondiffracting beams In the framework of GLMT and in spherical coordinates, the electric and magnetic fields E and H of an arbitrary shaped beam, with a time-dependence of the form expit , can be expanded in terms of VSWFs [1-4]. In particular, the radial components of the electric and magnetic fields play an important role in this framework because they allow one to determine the BSCs. These radial components are given by
Er E0
n
C
pw n
g nm,TM
n 1 m n
nn 1 m j n kr Pn cos expim , r
(9.a)
nn 1 m j n kr Pn cos exp im , r
(9.b)
and
Hr H0
n
C
n 1 m n
pw n
g nm,TE
7
where jn kr is the first kind spherical Bessel function, k is the wavenumber and Pnm cos is the associated Legendre function given by [61]
Pn
m
m cos 1m sin m d Pn cosm , d cos
(9.c)
and Cnpw (with pw standing for “plane wave”) denotes expansion coefficients which appear in the Bromwich formulation of the classical Lorenz Mie theory [2] according to
C
pw n
n 1 i
k
2n 1 . nn 1
(9.d)
In Eq. (9.a-b), BSCs may be expressed in terms of radial electric and magnetic fields components by using quadrature methods [62] i.e g nm,TM i n 1 n m ! kr 2 E r kr / E 0 m im sin dd . m H kr / H Pn cos e 0 g n ,TE 4 n m ! j n kr 0 0 r
(10)
By inserting Eqs.(8.a-b) into Eq.(10), BSCs of nondiffracting beams can easily be deduced as
ND m n ,TM ND m
g g n ,TE
ND m C l g n,TM l C ND g m l n ,TE l
where g nm,TM
BB
l
and g nm,TE
BB
l
BB
l
BB
l
,
(11)
are BSCs of HOBBs.
The use of this result depends on our ability to calculate the coefficients ClND and the BSCs
g
BB m n ,TM l
and g nm,TE
BB
l
. Calculations of the first kind ClND of coefficients for fundamental
nondiffracting beams will be given in the sequel. For the second kind of coefficients, namely BSCs of BBs, Mitri in [63] calculated them numerically by using double quadratures. For analytical calculations, Taylor and Love [52] used the angular spectrum decomposition to derive BSCs for zeroth-order Bessel beams. The same procedure was used by Lock [14] to analyze BSCs of zeroth-order Bessel beams and by Chen et al [53] to determine BSCs of polarized BBs of arbitrary order. Recently, Wang et al [64] derived a general description of higher order Bessel beams, which take into account results concerning Davis Bessel beams
8
type. According to this work and [16], TE and TM BSCs for circularly symmetric BB are given by gm n , TM TE
BB
m m 1 g 1m m / 2 n m ! e ik z z0 i l m 1 e i l m 1 0 J l m 1 k t 0 n cos m n cos i n m ! l 1 i l m 1e i l m 1 0 J l m 1 k t 0 nm cos m nm cos , 1
(12.a)
where nm and nm are the generalized Legendre functions given by
nm cos
Pnm cos , sin
(12.b)
dPnm cos . d
(12.c)
and
nm cos
In Eq. (12.a), g 1 cos / 4 is introduced to reduce BBs to Davis Bessel beams used in [16, 65]. 0 , 0 , z0 are the cylindrical coordinates of the beam center in the used coordinate system. is the axicon angle mentioned above in geometrical interpretation of Whittaker integrals. In Fig.1, we reproduce plots of Fig. (2.a-b) of [16] of BSCs of second order BBs for m=1,3. It should be noted that for an on axis BBs of l-order all BSCs are zero except for
m l 1 , corresponding to m 1 and m 3 for l 2 . Thus, in the series of Eq. (11) and for a given value of m only terms corresponding to l m 1 are different from zero. BSCs of any
nondiffracting beam are then obtained by multiplying g nm,TM
m l 1 , we have g nm,TM
BB
m 1
i g nm,TE
BB
m 1
BB
m 1
by CmND1 . Furthermore, for
. In the sequel, this example of BBs will be used to
calculate BSCs for other fundamental nondiffracting beams (plane wave, cosine, Mathieu and parabolic beams). But the application of Eq. (11) for these beams depends on the calculation of overlap coefficients ClND and on the convergence of series in Eq. (5.a), Eqs. (8.a-b) and Eq. (11) (for more discussion on convergence of BSCs of BBs see [16] and [32, 51] for the convergence of the series in Eq. (11)).
9
(a)
Figure 1: g nm,TM
(b)
BB
l
of BBs for an axicon angle 10 , 20° and 30°:
(a) l 2, m 1 ; (b) l 2, m 3 .
3.1 Plane wave and cosine beams Homogeneous plane waves are solutions of HE in Cartesian coordinates. The electric field and the angular spectrum associated with these beams are, respectively
U PW x, y U 0 expik x x expik y y ,
(13.a)
and
APW ,
(13.b)
where is a constant angle. In this case k x kt cos and k y k t sin . Some patterns can be
obtained
from
superposition
of
plane
waves.
For
example,
if
we
take
ACB , we obtain cosine beams, introduced by Jiang in Ref. [29]. 2 2 The electric field of these beams is
U CB x, y U 0 exp ik y y exp ik y y 2U 0 cosk y y .
10
(13.c)
Two-dimensional intensity distribution of these beams is given in Fig. 2. The overlap coefficients between plane wave or cosine beam and Bessel beams can be obtained by inserting the angular spectrum of these beams in Eq. (5.b). ClPW and ClCB are then given by
ClPW
1 exp il , 2
(13.d)
2 cos l . 2
(13.e)
and
ClCB
Since C lCB C CBl , Eq. (5.a) can be written as
U CB r 2 0,l ClCB (i ) l 2 J l k t cos(l ) .
(13.f)
l 0
Fig. (2.b) displays a comparison between the formulation of scalar Cosine beam using Eq.(13.c) and the expansion formulation in terms of Bessel beams basis by using the coefficient ClCB (Eq.(13.f)), From this figure, we observe a good agreement between the two
formulations of cosine beams. In Fig (2.c), we plot g1n,TM
CB
versus n, where g1n,TM
CB
are
calculated by expanding Cosine beams in terms of circularly symmetric BBs such as described in [16].
11
(a)
(b)
(c) Figure 2: Cosine beams: (a) Two dimensional intensity distribution of a cosine beam with k y L 5 (the size of transverse display area is L L ); (b) Comparison between intensity of a cosine beam using formulation of Eq. (13.c) and BB expansion of Eq. (13.f); (c) BSCs performed with Eq. (11) for axicon angles 10 , 20° and 30°.
3.2 Higher order Bessel Beams For a m-th order BBs, the angular spectrum is [50, 66]:
A BB
1 2
expim .
(14.a)
In this case the overlap coefficients are given by
12
ClBB
1 2
2
expim exp il d 0
lm
0 if m l , 1 if m l
(14.b)
where lm is the Kronecker index. BSCs of these beams are well known from various works as [14,16,52,53,63].
3.3 Mathieu beams In elliptic cylindrical coordinates, the transverse HE can be split in two differential equations: the radial Mathieu equation and the angular Mathieu equation. Solutions of these equations are radial and angular Mathieu functions, respectively. In terms of these functions, the even and odd fields distributions are given by
U e r U 0e Jem , q cem , q ; m=0,1,2 … for even modes,
(15.a)
and U o r U 0 o Jo m , q se m , q ; m=1,2, … for odd modes,
(15.b)
where Jem and Jom are mth-order even and odd radial Mathieu functions, respectively. cem and so m are mth-order even and odd angular Mathieu functions, respectively.
q
h 2 k t2 is the ellipticity parameter and h is the semi-confocal parameter. , denote 2
transverse
elliptic
coordinates
related
to
the
transverse
coordinates
x, y
by
x h cosh cos and y h sinh sin . The angular spectra of Mathieu beams are the
angular Mathieu function cem ( , q) for even modes and sem ( , q) for odd modes [31, 32, 50, 66], i.e MB Ae AoMB
1
1
cem ( , q ) sem ( , q )
1
1
A q cos j (m) j
j 0
.
(15.c)
B q sin j (m) j
j 1
A(jm ) q and B (jm ) q are the expansion coefficients of Mathieu functions. By inserting
AeMB and AOMB in Eq. (5.b), the coefficients ClND corresponding to Mathieu beams modes are given by 13
CeMB ,l
A(j m) if l j 0 2 A( m) j if l j 0 2 2 otherwise 0
CoMB ,l
B (j m) if l j i 2 0 otherwise
for even modes,
(15.d)
for odd modes.
(15.e)
By using these coefficients one can expand the field and the BSCs of Mathieu beams in terms of BBs and BSCs of these later beams, respectively. Using these ClMD , Mathieu beams fields can be written as:
U eMB r 2 A (j m ) (i ) l J l k t cosl for even modes,
(15.f)
j 0
and
U oMB r 2 B (j m) (i) l J l k t sin l for even modes.
(15.g)
j 1
The above results was already used in Ref. [32] for studying paraxial propagation of Mathieu beams and in Ref. [20], for evaluating the validity of localized integral
approximation in the case of zeroth-order Mathieu beams. In Fig. (3), we present g nm,TM
MB
versus n from Eq. (11), for zeroth order Mathieu beams and fourth order even Mathieu beams. From these figures, it can be seen that the BSCs are very well defined for n ranging from 0 to 150.
14
(a)
(b)
Figure 3: Evolution of BSCs versus n of Mathieu beams for q=20, m=1, axicon angles
10 , 20° and 30°: (a) zeroth-order Mathieu beams; (b) Fourth order even Mathieu beams.
3.4 Parabolic beams or Weber beams In literature, there are various representations of transverse separable solutions of HE expressed in cylindrical parabolic coordinates. In [33], Bandres et al. presented the solution as a product of parabolic cylinder functions Pe and Po , which are two series of parabolic cylinder coordinates. In Refs. [50, 51, 54, 61, 67] transverse solutions are connected with Weber differential equation whose solutions are expressed with parabolic cylinder functions D . In the work of Rodríguez-Lara [51], the expression of parabolic beams is given by 2
U ePB a , r U 0e
1 ia 4 2 U e a, u Ve a, v 2 sin
even mod es ,
(16.a)
odd mod es ,
(16.b)
and 2
U oPB a, r U 0o
3 ia 2 4 2 U O a, u VO a , v sin
where u, v are parabolic coordinates related to Cartesian coordinates by u iv 2 x iy , a 1/ 2
is a separation constant. U e ,o a, u and Ve,o a, u are given by
15
1 ia 1 U e a, u exp ik t u 2 / 2 1 F1 , , ik t u 2 , 4 2 2
(16.c)
1 ia 1 Ve a, v exp ik t v 2 / 2 1F1 , , ik t v 2 , 4 2 2
(16.d)
3 ia 3 U e a, u 2k t u exp ik t u 2 / 2 1 F1 , , ik t u 2 , 4 2 2
(16.e)
and 3 ia 3 Ve a, v 2k t v exp ik t v 2 / 2 1F1 , , ik t v 2 , 4 2 2
(16.f)
where 1 F1 is a Hypergeometric function. In [50, 66, 67], the angular spectra arising from the plane wave decomposition of even and odd modes of parabolic beams are
exp ia ln(tan ) 2 , AePB a, 2 sin
(16.g)
and
AoPB a, i sgn AePB a, .
(16.h)
Even and odd solutions are constructed from the sum or the difference of fields obtained from the Whittaker integral. By inserting the above angular spectra in Eq. (5.b), the overlap coefficients corresponding to parabolic beams can be written as
C ePB ,l
1 2
a 1 i 2 4
1 cos
a 1
1 cos i 2 4 exp(il )d ,
(17.a)
and
CoPB,l
a 1 a 1 i i i 1 cos 2 4 1 cos 2 4 exp(il )d 2 0 0
a 1 i 2 4
1 cos
a 1 i 2 4
1 cos
exp(il )d
16
.
(17.b)
By using results from [51], and result 3.892. 3 p .486 of [61], overlap coefficients are given by (here we use the same notation as in [51]) C ePB ,l
1 2 a i ) / 4 C l , a C l , a , e 2
C oPB,l
(18.a)
i 2 a i ) / 4 e C l , a C l , a , 2
(18.b)
with 1 l C l , a l 1 f l , a i sgn a f l ,a , 2
(18.c)
and
1 2 ia ia 2 F 1 ia , 1 ia;1 l ia ; 1 , f l , a 2 1 1 l ia 2 2 2
(18.d)
where 2 F1 denotes a Hypergometric function. In [51], Rodríguez-Lara demonstrated that the contribution of overlap coefficients increases as l increases until a given l and afterward it oscillates around a maximum value of these coefficients. This structural property permits to conclude that it is not feasible to efficiently construct a parabolic beam through a finite superposition of just a few Bessel beams. Recently, Khonina et al [68-69] expanded the angular spectra function of the even modes with an additional index p (integer or fractional) in Fourier series as 1 2 2 AePB a, a le a 1 0 ,s cos s bse a cos s , 2 T T s 0
(19.a)
where as , bs are Fourier coefficients defined by T/2
a se a
2 2 AePB d , , o a, cos s T T / 2 T
(19.b)
and T /2
bse a
2 2 AePB d . , o a , sin s T T / 2 T
(19.c)
The period of the function AePB , o a, is T 2 . Relying on the use of the additional index p, Khonina et al. distinguished between the case a 0 and the case a 0 to develop this beam 17
in terms of Bessel beams. For p 1 and for a 0 , the authors demonstrated in [68] that bse 0 0 and
a se 0
1s
2 . 3 3 s s 4 4
(19.d)
Substituting the Fourier decomposition in Eq. (4.c), we deduce
U ePB a, r U 0e exp ikz 2 0, s i a se 0 coss J s k t . s
(19.e)
s0
For the case a 0 , the obtained integral expressions for a se a and bse a have to be calculated numerically, insofar as it is difficult to reduce them to any closed formula. Furthermore, even if we calculate a se and bse the Fourier series cannot be used in this case because of their slow convergence as for Lara approach [51]. In the following, we therefore contented ourselves to only study the case a 0 . Fig. (3) confirms that, in the case a 0 , the series of Eq. (19.e) converge and BSCs can be calculated from those of BB. By inserting AePB a 0, into Eq. (5.b), the overlap coefficients are given by
C
PB l
e 1 a s (0)1 0, s coss exp il d 2 s 0 2 1
.
e 1 a 0 1 if l s 2 l 2 0,l 0 otherwise
(20)
With these overlap coefficients we easily deduce the expansion series in Eq. (19.e) of parabolic beams in terms of BBs expressed from Eq. (6). For parabolic beams odd modes, we use AoPB a, of Eq. (16.g) to derive similar overlap coefficients. In Fig. (4), we plot
parabolic beams for a 0 and its g nm,TM
PB
versus n calculated from Eq. (11).
18
(a)
(b)
(c) Figure 4: Parabolic beam when a 0 : (a) Intensity of beam computed by the exact solution (Eq.4.c); (b) Intensity of beam computed by the finite series solution (Eq.(19.e); (c) Evolution of BSCs versus n for m=1 and for axicon angles 10 , 20° and 30°.
From these Figures, we deduce that the Bessel expansion gives the same results as the numerical calculation of the intensity from Whittaker integrals. Curves in Fig. (4.c) are validation of Bessel expansion for BSCs. 4. Conclusion In this work, we have introduced a new method of calculating the BSCs used in GLMT for fundamental scalar or vector nondiffracting beams. We have exploited the relationship between separable exacts solutions of Helmholtz equation. Theoretical study shows that any exact solution can be expressed as a sum or integral over basis functions formed by other 19
exact solutions. Whittaker integrals express any exact solutions of Helmholtz equation in terms of plane waves. In this paper, we have expressed exact solutions of Helmholtz equation i.e nondiffracting beams as sum of BBs of various orders. The expansion coefficients are named overlap coefficients which are calculated from angular spectra of nondiffracting beams and BBs. The overlap coefficients are used to expand BSCs of nondiffracting beams in terms of BSCs of BBs. The proposed method can be used for all conventional nondiffracting beams except for parabolic beams for a 0 , because of the slow convergence of the expansion series. As application of this method, we have used BSCs of circularly symmetric BB to derive BSCs of plane wave, cosine, Mathieu and parabolic beams ( a 0 ) and plot the evolution of these BSCs versus n. Finally, the proposed method can enrich the possibility of employing the GLMT in studying various optical applications that use nondiffracting beams.
20
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