Spherical evanescent waves of scalar dipole

Accepted Manuscript Title: Spherical evanescent waves of scalar dipole Author: A. Suhadolnik PII: DOI: Reference:

S0030-4026(17)30200-0 http://dx.doi.org/doi:10.1016/j.ijleo.2017.02.057 IJLEO 58873

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28-11-2016 14-2-2017

Please cite this article as: A.Suhadolnik, Spherical evanescent waves of scalar dipole, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2017.02.057 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Spherical evanescent waves of scalar dipole A. Suhadolnik University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, 1000 Ljubljana, Slovenia

ABSTRACT A scalar dipole has a propagating field and evanescent field with the exponential decay. In this paper, the exact calculation of both fields has been performed in a zero frequency limit. In this limit the evanescent dipole field decays according to the power law around a wave source. The calculation bases on an angular spectrum representation using Weyl’s expansion technique which enables separation of the evanescent and propagating field.

Keywords: Spherical wave; evanescent wave; propagating wave; Weyl’s expansion.

E-Mail: [email protected].

1. Introduction It is well known fact that an exponential decay evanescent waves appears outside the spherical source [1,2]. Recently a zero frequency limit Coulomb [3,4] field from a scalar source point was described. The exact calculation of the evanescent and propagating field of the zero frequency scalar point source has been performed elsewhere [5]. It was found that the evanescent field power law decay prevails in this case. In this paper the evanescent zero frequency field of the scalar dipole field is described. The evanescent and propagating ( homogeneous) field was calculated by using Weyl’s expansion technique [6].

2. Spherical waves Waves inside and outside the sphere can be calculated with the Helmholtz differential equation [7]

2   k 2   0,

(1)

where Ψ is a scalar function and k wave number. The wave number k   / c is determined with a wave velocity c and angular frequency

 . The solutions of the Helmholtz equation for outgoing waves in a spherical coordinate system are spherical Hankel functions of the first kind of order n, namely hn(1) (kr ) . A zero order n  0 spherical Hankel function represents the outgoing wave of the scalar point source [8] h0(1) (kr ) 

eikr , ikr

(2)

2

where r 2  x 2  y 2  z 2 .

3. Weyl’s representation In order to represent spherical waves as plane waves, an angular spectrum representation is used. Only the half space z  0 will be considered in the calculations while the other half space z  0 can be calculated similarly. The angular spectrum representation transforms the spherical waves into the planar waves. By using Weyl’s integral formula one can represent the spherical wave as a sum of the plane waves [9]  

eikr 1 1 i ( kx x  k y y  kz z )  dk x dk y . kz e   ikr 2 k  

(3)

The complete field G  eikr / r is the sum of a propagating field G p where k z  k 2  k x 2  k y 2 if k x2  k y2  k 2 and evanescent or decaying field Ge where k z  i k x 2  k y 2  k 2 if k x2  k y2  k 2

G  Gp  Ge .

(4)

According to integral (3), the evanescent scalar field is equal Ge ( x, y, z ) 

i 2



1 k x2  k y2  k 2 k z

e

i ( kx x  k y y  kz z )

dk x dk y , k z  i k x 2  k y 2  k 2 .

(5)

The propagating scalar spherical field is then Gp ( x, y, z ) 

i 2



1 k x2  k y2  k 2 k z

e

i ( kx x  k y y  kz z )

dk x dk y , k z  k 2  k x 2  k y 2 .

(6)

The solution of the above integral (5) gives the expression for the evanescent scalar field of the point source [10, 11] 1 Ge ( x, y, z )  2U 0 (kr  kz, k x 2  y 2 )  J 0 (k x 2  y 2 )  .  r

(7)

Similarly (6) gives 1 Gp ( x, y, z )  eikr  J 0 (k x 2  y 2 )  2U 0 (kr  kz, k x 2  y 2 )  ,  r

(8)

where J0 and U0 are the zero order Bessel function and zero order Lommel function of two variables respectively [12] J 0 (k x 2  y 2 ) 

1 2



 e

 ik x 2  y 2 sin



d ,

(9)

 kr  kz U 0 (kr  kz, k x  y )    1   k x2  y 2 m 0  

2

m

2

   

2m

J 2 m (k x 2  y 2 ).

(10)

The Bessel function of order 2m is defined as

J 2 m (k x 2  y 2 ) 

1 2



 e 

i (2 m  k x 2  y 2 sin )

d .

The dipole scalar field can be calculated as the derivative of G

(11)

3

G '( x, y, z ) 

d (eikr / r ) z  3 (ikr  1)eikr , dz r

(12)

and consequently the evanescent dipole scalar field is the derivative of Ge . The partial derivative of the Bessel function is

J 0 (k x 2  y 2 ) / z  0 . The partial derivative of the Lommel function of two variables is equal 2U 0 ( , ) / u  ( /  )2  1 U1 ( , )  ( /  ) J1 ( ) and inserting   kr  kz and   k x 2  y 2 into this equation gives from (7) the expression for the derivative Ge '( x, y, z ) .  1  (2U 0 (kr  kz , k x 2  y 2 )  J 0 (k x 2  y 2 ))   z  r  z  k 2kz 2 2 2 2  2 2  3 J 0 (k x  y )  2U 0 (kr  kz , k x  y )  2 x  y J1 (k x 2  y 2 )  2 U1 (kr  kz , k x 2  y 2 ).  r r  r

Ge '( x, y, z ) 

(13)

Considering equation (8) the dipole propagating scalar field is then equal

G p '( x, y, z )  

z  k (ikr  1)eikr  J 0 (k x 2  y 2 )  2U 0 (kr  kz, k x 2  y 2 )   2 3   r r

x 2  y 2 J1 ( k x 2  y 2 ) 

2kz U1 (kr  kz, k x 2  y 2 ). r2

(14)

4. Special solutions Both field equations can be simplified in two special directions. The first special direction involves the case where x  y  0; r  z . To simplify both equations, property of Lommel function U0 ( , )  12 J 0 ( )  cos , U1 ( , )  12 sin  has to be applied. In case of

x  y  0; r  z , the Lommel function is U 0 (kr  kz, k x 2  y 2 )  U 0 (0,0)  1 . The Bessel function is in this case J 0 (k x 2  y 2 )  J 0 (0)  1 and both fields are simplified [13-17] Ge '(0, 0, z )   G p '(0, 0, z ) 

1 , z2

(15)

ikeikz 2ieikz sin(kz / 2)  . z z2

(16)

The second exceptional case is xy plane direction where z  0 . In this case r  x 2  y 2 and the evanescent and propagating scalar dipole fields are according to (13) and (14) by using the similar calculation as in previous case

Ge '( x, y, 0) 

kJ1 (k x 2  y 2 ) x2  y 2

,

(17)

4

G p '( x, y, 0)  

kJ1 (k x 2  y 2 ) x2  y 2

(18)

.

These special cases are only consequence of the mathematical calculations and have no physical background.

5. Zero frequency limit of scalar dipole In the static limit when k  0 or c   the evanescent scalar dipole field from the equation (13) is Ge '   z / r 3 . In this case the complete scalar dipole field has the same value G '  z (ikr  1)eikr / r 3   z / r 3 , and consequently the propagating scalar dipole field is

G p '  0 . This means that the propagating scalar dipole field vanishes in the static limit and the evanescent dipole scalar field has power law decay, so the double integral (5) is equal Ge '( x, y, z; k  0) 

1 2



k x2  k y2  k 2

e

2 2 i ( kx x  k y y )  kx  k y z

e

dk x dk y  

z , r3

(19)

and similarly (6) Gp '( x, y, z; k  0) 

1 2



k x2  k y2  k 2

1 i ( kx x  k y y ) i e e i

kx2  k y 2 z

dk x dk y  0.

(20)

This limit is of course only hypothetical, because the ultimate velocity is limited to the velocity of the light and cannot be unlimited. In the zero frequency limit only the decaying evanescent scalar dipole field spreads around the source without the propagating scalar dipole field.

6. Conclusions In summary, the special cases of the spherical dipole wave propagation are discussed. It is shown by the angular spectrum representation using Weyl’s integration that the evanescent field of the dipole could decay by the power law if the angular frequency of wave approaches to the zero value. In this case the propagating part of the wave field drops to the zero and only the evanescent field exist. In other cases the evanescent field has exponential decay except in two special directions which are consequence of mathematical calculations and have no physical explanation. These findings can be useful by the calculation of the spherical wave sources.

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