Optical radiation force circular dichroism spectroscopy

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Optical radiation force circular dichroism spectroscopy F.G. Mitri PII: DOI: Reference:

S0022-4073(19)30982-3 https://doi.org/10.1016/j.jqsrt.2020.106850 JQSRT 106850

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Journal of Quantitative Spectroscopy & Radiative Transfer

Received date: Revised date: Accepted date:

12 December 2019 20 January 2020 21 January 2020

Please cite this article as: F.G. Mitri , Optical radiation force circular dichroism spectroscopy, Journal of Quantitative Spectroscopy & Radiative Transfer (2020), doi: https://doi.org/10.1016/j.jqsrt.2020.106850

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Optical radiation force circular dichroism spectroscopy F.G. Mitria) Santa Fe, New Mexico 87508, USA (Submitted 11 December 2019, revised 20 January 2020)

Highlights - This work introduces the optical radiation force circular dichroism spectroscopy method - This method is defined as the force difference of left-handed and right-handed circularly polarized waves - The example of a perfect electromagnetic circular cylinder is considered - A line source of cylindrically diverging waves with circular polarization is assumed - Radiation force and energy efficiency results are computed and discussed

Abstract This work introduces the method of circular dichroism spectroscopy in the framework of the electromagnetic/optical radiation force theory. This analytical tool is defined here as the difference in radiation force of left-handed and right-handed circularly polarized electromagnetic waves illuminating an object exhibiting rotary polarization. The example of a lossless material, such as the perfect electromagnetic conductor (PEMC) cylinder having a circular geometric cross-section, is considered. The modal expansion method in cylindrical coordinates is used to obtain exact mathematical series expansions for the longitudinal radiation force per-length (i.e. acting along the direction of wave propagation) considering left-handed and right-handed circularly polarized cylindrically diverging waves emanating from a line source. The case of plane progressive waves is recovered when the source is located far from the cylinder. Numerical illustrative results for the dimensionless radiation force functions as well as the scattering, extinction and absorption energy efficiencies and their co-polarized and cross-polarized components are performed with particular emphasis on the size parameter of the cylinder, the dimensionless distance parameter from the line source, and the admittance parameter of the cylinder. The results reveal that the individual radiation force functions for left-handed and right-handed circularly polarized waves can be negative, zero, or positive depending on the cylinder distance from the source. Moreover, the optical radiation force circular dichroism (ORFCD) and the extinction energy efficiency circular dichroism (EEECD) are positive for a negative admittance of the cylinder, while they reverse sign for a positive admittance. While the EEECD shows some form of symmetry versus admittance sign change, the ORFCD does not. The possibility of achieving invisibility cloaking for a small PEMC cylinder is also investigated. The present ORFCD spectroscopy method is applicable to any cylinder material exhibiting rotary polarization such as chiral, topological insulator, plasma, liquid crystal etc. Keywords: Electromagnetic scattering; electromagnetic radiation force; circular dichroism/rotary polarization; PEMC cylinder; circularly polarized waves; multipole expansion method

1. Introduction

[17], stopped-flow [18] and synchrotron radiation [19] spectroscopies. In all these methods the difference of energy extinction (by absorption and scattering [20]) between lefthanded and right-handed circularly polarized wavefields is quantified. As such, the extinction energy efficiency circular dichroism (EEECD) spectra allow studying the shapes of particulate matter and arrangement of atoms [21]. From a theoretical standpoint that could improve structural analyses of unconventional materials by means of radiation forces and scattering of EM/optical waves, it is of some importance to develop a novel spectroscopy method that would complement the state-of-the-art circular dichroismbased techniques available to date. Such an analytical spectroscopy method may be utilized in the realm of optical tweezers and particle manipulation, entailing the subtraction of radiation force spectra of left-handed and right-handed circularly polarized EM/optical waves illuminating an object

Circular dichroism [1] is an interesting property exhibited by some type of materials possessing internal asymmetry that absorb and scatter electromagnetic (EM) and optical radiation based on polarization. During this process, a rotation of the plane of polarization [2-5] (known also as depolarization [6]) is induced, which offers a quantitative tool to characterize particles via spectroscopy. For example, typical information related to the detailed inner structure of proteins, liquid crystals, nucleic acid, nanostructures etc., can be obtained [7]. Previously, such materials have been defined as “optically active” [8] to describe this property. However, this description must be distinguished from active materials containing active sources which radiate EM/optical/acoustical waves [9-12]. Numerous circular dichroism-based techniques exist [13, 14], encompassing electronic [15], magnetic [16], vibrational 1

JQSRT (2020) rly polarized field affects the radiation force by a PEMC cylinder material, since the incident field is a superposition of two dephased TM and TE modes. This topic is relevant in understanding the fundamentals of EM radiation force theory for particles exhibiting rotary polarization (or TM ⇆ TE mode conversion). Moreover, applications in the realm of particle manipulation, stabilization and optical tweezers have considered circularly polarized waves because of their intrinsic properties. The purpose of this analysis is therefore focused on developing an exact analytical formalism for the modeling for the EM/optical radiation force on a circular PEMC cylinder in both left-handed and right-handed circularly polarized field, emanating from a line source of cylindrically diverging waves (see panels (a) and (b) of Fig. 1). The analytical formalism presented in the following is exact, without any approximations related to particle size, such that the Rayleigh, Mie and geometrical optics regimes can be all considered. As such, the ORFCD spectroscopy method can be introduced. In the following, section 2 presents a complete analysis of the scattering used to derive the exact expressions of the radiation force functions and scattering, extinction and absorption energy efficiencies with emphasis on the crosspolarized waves. In section 3, numerical simulations are considered and discussed. Verification and validation stemming from the law of energy conservation applied to scattering are also performed. Finally, section 4 presents the conclusion of this work.

2. Method Consider a line source emitting an electromagnetic (EM) field composed of either left-handed or right-handed circularly polarized cylindrical diverging waves, denoted respectively by LHCP and RHCP (see panels (a) and (b) of Fig. 1). The resulting incident electric field vector is expressed as the combination of a pair (i.e., TE and TM) of linearly polarized components as

Fig. 1. Graphical representation for the interaction of an optical/EM lefthanded circularly polarized field of cylindrical diverging waves emanating from a line source, incident upon an infinitely-long PEMC cylinder having a circular cross-section of radius a. The incident field propagates along the xaxis, perpendicularly to the cylinder z-axis.

allowing rotary polarization. This method is defined here as optical radiation force circular dichroism (ORFCD) spectroscopy. In this paper, the aim is to introduce the ORFCD from a theoretical perspective, stemming from the analysis of the EM/optical scattering by a perfect electromagnetic conductor (PEMC) [22, 23] lossless cylinder [24] chosen as an example. The PEMC material can induce mode conversion (i.e. optical rotation or depolarization) such that both co-polarized and cross-polarized waves are generated in the scattered field [2433]. Other materials such as chiral, plasma and topological insulators exhibit similar properties. As such, the crosspolarized field contributes significantly to the radiation force experienced by the object as shown in the recent works [3436], in which linearly polarized transverse magnetic (TM) wave incidence has been considered. The challenge that remains to be addressed is how a circula

LHCP H 1  kR  cyl . div . RHCP Einc  E0e  it  e  ie z  01 , H 0  kr0 

(1)

where the superscripts LHCP and RHCP correspond to the (+) and (–) signs in the right-hand side of Eq.(1), respectively. E0 is the electric field amplitude and (r,, z) form a cylindrical coordinates system centered on the cylinder as shown in Fig. 1. The parameters e and e z denote the unit vectors along the polar and axial directions, respectively. As shown from

e  it is considered, 1 but omitted from the field equations for convenience. H 0 . Eq.(1), a time-dependence in the form of

is the cylindrical Hankel function of the first kind of order zero, k = 1/2/c is the wavenumber in the host medium,  is the dielectric permittivity,  is the angular frequency and c is

Electronic mail: [email protected]

2

JQSRT (2020) the speed of light. R is the distance from the center of the line source to an observation point in the transverse plane (xy), and r0 is the distance between the source and the center of the cylinder. The incident waves impinge perpendicularly on the z-axis of an infinitely long cylinder of radius a made of a PEMC material. The medium of wave propagation is homogeneous, non-dissipative and non-magnetic. Application of Graf’s addition theorem (pp. 142-144 in [37]) for cylindrical wave functions (Eq. (2) on p. 360 in [38]), the modal expansion method [39, 40] is used to express Eq.(1) as a mathematical series as,

LHCP cyl . div . RHCP Einc  r, r0 , 

 r0  r a



  1 H



1

n

n

n

n 0

LHCP cyl . div . RHCP Hsca  r, r0 ,  r0  r  a 





where H n1   is the cylindrical Hankel function of the first kind of order n, H n1 '  is its derivative with respect to the

E0 H 0  kr0  1

argument. The coefficients

TM TE

 kr0    J n  kr  e  iJ n  kr  e z  cos  n of, polarization, respectively, whereas Cn

 i  r0  r a



  1 H



n

n

n 0

1 n

 kr0  



E0 H 0  kr0 







n 0

H n   ka  CnTM TM 1

iJ n  kr  e  J n  kr  e z  cos  n .  '

'



 iM



 H n1 '  ka  CnTM TE   J n'  ka  ,

(7)

H n   ka  CnTE TE 1



 iM



 H n1  ka  CnTE TM   J n  ka  ,

(8)

H n   ka  CnTE TE 1



'



 i  M H n1 '  ka  CnTE TM   J n'  ka  ,

E0 H 0  kr0  1

   CnTM TE  CnTE TE  H n1 '  kr  e   cos  n ,  n  1 H n  kr0    i  CnTM TM  CnTE TM  H n1  kr  e z    n

(6)

1

conversion in the scattered waves (denoted by TM ⇆ TE), an incident circularly polarized field scattered from its surface would enhance the generation of TM and TE mode conversion. This intrinsic property stems from the fact that circular polarization consists of combining two linear TM and TE components that are perpendicular to each other, equal in amplitude, but have a phase difference of π/2. Consequently, the total electric and magnetic vector fields consist of all contributions, including the cross-terms (TM  TE and TE  TM). The scattered electric and magnetic vector fields expressions are given as,

r0  r  a



 i  M H n1  ka  CnTM TE   J n  ka  ,

Since the PEMC cylinder material induces EM mode



TE TM

and Cn

1

(3)

LHCP cyl . div . RHCP Esca  r, r0 , 

CnTE TE

H n   ka  CnTM TM

Based on Maxwell’s equations [41], the incident magnetic vector field is obtained from Eq.(2) as,

 r, r0 , 

and

describe those of the scattered cross-polarized waves due to mode conversion. Applying the appropriate boundary conditions for the PEMC cylinder, such that the tangential field components vanish (see Eq.(9) in [24]), the following system of linear equations is obtained as,

' kind and J n   is its derivative with respect to the argument.



CnTM TM

correspond to the scattered co-polarized waves for each type

'

function, J n   is the cylindrical Bessel function of the first

 cyl . div . Hinc

 i  CnTM TM  CnTE TM  H n1 '  kr  e   cos  n ,    CnTM TE  CnTE TE  H n1  kr  e z   

n

(5)

 n   2   n 0  and ij is the Kronecker delta

LHCP RHCP

E0 H 01  kr0 

 n  1 H n1  kr0  

n 0

(2) where

 i 

1

(4) 3

(9)

where M is known as the admittance [24] of the PEMC cylinder (i.e., a measure of electromagnetic conduction of the cylindrical material). Based on Eqs.(6)-(9), the scattering coefficients for the copolarized and cross-polarized scattered waves are determined as,

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



H n   ka 

 J n  ka   i  M H n1  ka  CnTM TM 

 J n'  ka  1

Hn

 ka 

H n   ka  1

'

 iM   H  '  ka  ,   i  M  H    ka   iM   H  '  ka  1 n

1 n

1

CnTM TE 

(10)

1 n

 J n  ka 

H n1 '  ka   J n'  ka 





H n1  ka 

 i  M H n1  ka 

H n1 '  ka 

 iM



 H n1 '  ka 

(continue on page 8)

4

,

(11)

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Fig. 2. Panels (a)-(f) display the plots for the dimensionless radiation force functions [given by Eqs.(19)-(22)] versus the size parameter ka for a PEMC cylinder having an EM admittance M = 0.5. Panels (g) and (h) display the ORFCD components for plane and cylindrically diverging waves, respectively. For each plot, the dimensionless distance source-cylinder kr0 is constant and has the value indicated near each curve. Notice that kr0 is always larger than ka, ensuring the physical validity of the results (i.e., the source is external to the cylinder). When kr0 , the case of plane progressive waves is obtained.

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Fig. 3. Panels (a)-(f) display the plots for the scattering, extinction and absorption energy efficiencies [given by Eqs.(26)-(36)] for a PEMC cylinder having an EM admittance M = 0.5. Panel (g) displays the EEECD given by Eq.(40) for cylindrically diverging waves. For each plot, the dimensionless distance source-cylinder kr0 is constant and has the value indicated near each curve. Notice that kr0 is always larger than ka, ensuring the physical validity of the results (i.e., the source is external to the cylinder). When kr0 , the case of plane progressive waves is obtained.

6

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Fig. 4. The same as in Fig. 2, but the EM admittance of the PEMC cylinder is M = +1.

7

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Fig. 5. The same as in Fig. 2, but the EM admittance of the PEMC cylinder is M = –1.

8

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 J n  ka  CnTE TE 

 J n'  ka  H n1  ka  H n   ka  1

'

 iM   H    ka    i  M  H   '  ka  ,  iM   H    ka  '   i  M  H    ka  1 n

1 n

1 n

CnTE TM 

H n1 '  ka 

0

(12)

cyl . div ., x

 iM   H    ka    i  M  H   '  ka  1 n



Ycyl.div ., x 

Knowing the mathematical expressions for the total (i.e., incident + scattered) electric and magnetic field components, the derivation for the longitudinal (i.e., along the x-axis) radiation force per-length and its dimensionless function (or efficiency) can be accomplished for each type of polarization LHCP or RHCP as [42, 43],





E

inc



(14)



, Hinc , E zinc , H zinc  ,

 ... denotes the real part of a complex number, the superscript * denotes a complex conjugate, the subscript  in Eq.(14) indicates that the far-field limits (i.e., kr  ) of the field components are used, dS  Lrd er , where L is the length of a cylindrical surface enclosing the cylinder, r is the radial distance to a point in the polar plane, and e r  cos e x  sin  e y , is the normal unit vector pointing outwardly to the surface of the cylinder, and

2 ka

 2     Y p. p. w., x   kr0 



  J  kr  J n

0

n 1

 kr0   Yn  kr0  Yn 1  kr0  

n 0

          TM TM   nTM  1  2 nTM TM    nTM TM 1  2 nTM   1 1      nTE1TE 1  2 nTE TE    nTE TE 1  2 nTE1TE     TE TM TE TM TE     nTM TE 1 2  nTM   1 2     1 n 1 n     nTE TM 1  2  nTE1TM   nTE1TM 1  2  nTE TM     TM     nTM TM  nTE1TM  nTE TM  nTM  1    TM TM TE TM  TM  n 1  nTE TM  nTM  1  2    n     TM TE TE TE  TE TE TM TE   n 1 n n 1    n  TE      nTM TE  nTE1TE  nTE TE  nTM  1 



LHCP cyl . div . RHCP  , Hinc   , z   

c LHCP FcylRHCP . div ., x 2aLI 0

  1    2  1  H 0  kr0     

CnTM TE  CnTE TM .



(16)

For the circular cylinder of radius a, the longitudinal dimensionless radiation force function (or efficiency) is defined as,

of a matrix. Eqs.(11) and (13) also demonstrate that

LHCP RHCP

F  ex.

(13)

1 n

  E inc* H sca  E sca H inc*  z ,  ,   z ,  ,   S  1  sca* sca  inc* sca F     E z , H  ,  E , H z , , 2c   sca inc* sca sca*   E , H z ,  E , H z ,  dS   

 1  n   1, n  0,  0  otherwise 

the longitudinal component of the force vector is obtained after algebraic manipulation and can be recast as an exact partial-wave series expansion, LHCP   F RHCP 

,

n 

(15)

where the mathematical symbol ... indicates the determinant

where Ecyl .div. inc

  cos  n  cos    cos  d   2 0 

 J n  ka 

H n1 '  ka   J n'  ka  H n1  ka 



1 n

H n   ka  1

2

(17) where

 nTM TM  CnTM TM ,

e x and e y are

 nTM TE  CnTM TE ,

the unit Cartesian vectors along the x-axis and y-axis, respectively. Using the property of the angular integral,

 nTE TE  CnTE TE ,

 nTM TM  CnTM TM ,

 nTM TE  CnTM TE ,

 nTE TE  CnTE TE ,

 nTE TM  CnTE TM ,

 nTE TM  CnTE TM , I 0   E0

2

2,

 ...

denotes the imaginary part of a complex number, and 9

  , 

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Y pLHCP . p . w ., x is the radiation force function for the case of a left-

TE YcylTE. div ., x 

handed circularly polarized field of plane progressive waves (p.p.w.), which is expressed as,

  1    2  1  H 0  kr0     

  2  Y p.p. w., x      ka 





TM TM n

TM   nTM   nTE TE   nTE1TE 1

n 0

  nTE TM   nTE1TM

 nTM TE

2  ka

TE  nTM 1

   TM TM TM TM  TM TM TM TM  n 1  n  n 1  n    TM TE TM TE   TM TE  TM TE  n 1 n n 1  n   2   nTE TE nTE1TE   nTE TE  nTE1TE  TE TM TE TM  TE TM TE TM  n 1  n  n 1   n   TM TE TE TE TM TM TE TM   n 1   n  n 1    2   n    TE TM  TM TM  TE TE  TM TE   n n 1 n    n 1

      .     

cyl . div ., x

TE TE



,   cyl . div ., x

 Ycyl .div., x  Ycyl .div., x  Y

,

Y



2  ka

(19)

 2  TM TM   Y p. p. w., x   kr0 

  J  kr  J n

0

n 1

0

n 1

 kr0   Yn  kr0  Yn1  kr0  

(21)





2 ka

 2  ,  Y p. p. w., x    kr0 



  J  kr  J n

0

n 1

 kr0   Yn  kr0  Yn1  kr0  

n 0

          TM TE TM TE TE    nTM TE 1 2  nTM  2 1      n n 1 1       nTE TM 1  2  nTE1TM   nTE1TM 1  2  nTE TM    ,    TM     nTM TM  nTE1TM  nTE TM  nTM  1     TM TM TE TM TM  nTE TM  nTM  n 1     n 1   2   TM TE TE TE  TE TE TM TE  n 1 n n 1     n TE       nTM TE  nTE1TE  nTE TE  nTM 1 

TM YcylTM.div ., x 



n

n 0

  1    2   1 H  kr0      0

where,

  1    2  1  H 0  kr0     

  J  kr  J

,   cyl . div ., x

For convenience, Eq.(17) can be re-expressed as the sum of three distinct contributions resulting from the co-polarized waves components (denoted by the superscripts TM  TM and TE  TE, respectively), and a cross-polarized factor one (denoted by the superscript  ) related to mode conversion. It follows that, TM TM



    nTE1TE 1  2 nTE TE    nTE TE 1  2 nTE1TE   , 

(18)

   Y

 2  TE TE   Y p. p. w., x   kr0 

(22) TM TM

and the expressions for Y p . p . w., x ,

 kr0   Yn  kr0  Yn 1  kr0  

n 0

 TM    nTM 1  2nTM TM    nTM TM 1  2 nTM1TM   , 1  (20)

TE Y pTE. p . w., x ,

appearing in Eqs.(20)-(22) [corresponding progressive waves (p.p.w)] are given as,

Y

TM TM p . p . w .,x

 2      ka 



, 

and

Y p. p.w., x

to

plane



 

TM TM n

TM   nTM 1

n 0

TM TM 2  nTM TM  nTM   nTM TM  nTM  , 1 1

(23)

10

JQSRT (2020)

Y

TE TE p . p . w., x

2  and

Y



 2      ka 

 





TE TE n

n 0

TE TE n 1

TE TE n





  nTE1TE

TE TE n

TE TE n 1

 ,



, 

TM TM TM TE TE TE TE TM Qsca  Qsca  Qsca  Qsca  Qsca  Qsca  ,

(24)

(31) TM TM ext

Q



,   p . p . w., x

 2      ka 



  

TE TM n

  nTE1TM

 nTM TE

n 0

    .    

Q

Q



n

n

2

n 0



TM TM 2 n



TM TM 2 n

Q

TM TE Qsca 



2 ka

n n

2

n 0



TM TE 2 n

TE TE Qsca 

TE TM sca

Q



, 

Qsca  

2 ka



2  ka 4 ka

4  ka

n

2

n

n

2

n

n 0 

 n 0



 n 0





TE TM 2 n

  nTE TM , (29)

2

2

2



n

n

2

n 0



n

n

n

n 

2

TE TM n

 nTM TE  (34)

2

TM TE n

,

n 0

ext

ext

         Q  Q Q  , ext

sca

ext

(35) (36)

H n   kr0  1

n 

H 01  kr0 

.

(37)

Notice that lim  n   1 , and the infinite plane wave kr0 



  nTE TE ,

(33)

and

,

TE TE 2 n

 n  nTE TE , 2

n

abs

result is recovered. Now that all the physical observables are defined in exact partial-wave series expressions, the radiation force circular dichroism (RFCD) efficiency for cylindrically diverging waves emanating from a line source is obtained from Eqs.(17) and (19) as (continue on page 12),

(28)



  nTM TE  nTE TE   nTM TM  nTE TM   TE TM  nTM TM  nTE TE  nTM TE   n

 n  n 



n 0









(32)



n 0

ext

(27) 

2

n 0





 n  nTM TM ,

n

  ,   TM TM TE TE Q Q Q Q  ,

  nTM TE , 2

2 ka

4  ka

(26) 



,   ext

The analysis is further extended to derive the expressions for the energy efficiencies (or cross-sections), such as the scattering, extinction and absorption efficiencies. Those quadratic observables are useful in the study of the scattering and radiation force since they validate/verify the results from the standpoint of energy conservation. Stemming from the previous exact formalisms [44] for the scattering by a PEMC [24] or a chiral [8] cylinder, series expansions for the scattering, extinction and absorption energy efficiency factors have been derived analytically and are given as follows, 

2  ka

TE TE ext

(25)

2  ka



TE  nTM 1

   TM TE TM TE  TM TE TM TE  n 1  n  n 1  n   TE TM TE TM TE TM TE TM  2  n  n 1  n  n 1      nTM TE  nTE1TE   nTM TM  nTE1TM    2  TE TM TM TM    nTE1TE  nTM TE      n 1  n

TM TM sca



2  ka

  nTM TE   nTM TM   nTE TE    ,    nTE TM  nTM TM   nTE TE     (30)

11

JQSRT (2020)

Fig. 6. Panels (a)-(f) display the plots for the dimensionless radiation force functions [given by Eqs.(19)-(22)] versus the dimensionless source-cylinder distance kr0 for a PEMC cylinder having an EM admittance M = 0.5. Panels (g) and (h) display the ORFCD components for plane and cylindrically diverging waves, respectively. For each plot, the dimensionless cylinder size is ka and has the value indicated near each curve. Notice that kr0 is always larger than ka, ensuring the physical validity of the results (i.e., the source is external to the cylinder).

12

JQSRT (2020)

Ycyl .div ., x  Ycyl .div ., x  Ycyl .div ., x

3. Numerical results and discussions

 Ycyl,.div ., x  Ycyl,.div ., x   1    2   1  H  kr0     0  

4 ka

 2    Y p. p. w., x   kr0 



  J  kr  J n

Eqs.(38) and (40) establish the core of the ORFCD and EEECD spectroscopy formalisms for any cylindrical particle allowing rotary polarization in the field of circularly polarized plane progressive waves. In this analysis, a non-absorptive PEMC cylinder with a circular cross-section is considered and the host medium is air with a dielectric constant   1. The radiation force efficiencies for the cylindrically diverging left-handed and right-handed circularly polarized waves as well as the energy efficiencies related to scattering, extinction and absorption are computed. A MATLAB numerical platform is utilized for the simulations. Emphasis is given on varying the size parameter ka of the cylinder, the dimensionless distance from the source kr0 and the EM admittance parameter M. The mathematical series representing the components of the longitudinal radiation force function and energy efficiencies are truncated according to a  convergence criterion that was enforced to ensure accuracy  , and negligible numerical error. The convergence criterion is

0

n 1

 kr0   Yn  kr0  Yn 1  kr0  

n 0

           nTE TM 1  2  nTE1TM    nTE1TM 1  2  nTE TM     TE TM TE TM TE    nTM TE 1  2  nTM   1  2   n1   1 n   TM    nTM TM  nTE1TM   nTE TM  nTM   1    TM TM TE TM  TM  n 1   nTE TM  nTM     n  1  2   TM TE TE TE   TE TE TM TE   n 1 n n 1    n  TE      nTM TE  nTE1TE   nTE TE  nTM  1 

chosen such that Nmax = round[ka + 4 3 ka + 55], leading to a negligible relative numerical error in the order of ~10–14. This maximum truncation order is larger than the one used previously for plane progressive waves [45] (i.e., Nmax = round[ka + 4.05 3 ka + 2] ) because near the source (i.e., kr0 ~ ka), adequate convergence requires adding a larger number of multipoles in the series. Moreover, in all the computations, the condition kr0 > ka has been enforced, suggesting that the source is always external to the cylinder core, which warrants the physical validity of the results. First, a PEMC cylinder is considered with an EM admittance factor M = 0.5. Panel (a) of Fig. 2 displays the results for the longitudinal radiation force function for the copolarized waves TM  TM given by Eq.(20) versus ka, for six different values of kr0 (= 1, 2, 4, 6, 10 and  ). In the

(38) where Y p . p . w., x is obtained from Eq.(18) [or Eq.(25)] as,

Y p. p. w., x  Y p. p. w., x  Y p. p. w., x  Y p. p,. w., x  Y p. p,. w., x  Y p. p. w., x  4      ka 

TM TM

Rayleigh regime for which ka  0.5, Ycyl . div ., x is maximal. As



 

ka increases and approaches kr0, it diminishes, vanishes, and turn negative as the source is moved close to the cylinder. When the source is moved to infinity, i.e., kr0  , the plane

TE   nTE1TM   nTM TE   nTM 1

TE TM n

n 0

 nTM TE  nTE1TE   nTM TM  nTE1TM  4  TE TM TM TM    nTE1TE  nTM TE    n 1  n

TM TM

.

progressive wave limit is reached, and Ycyl . div ., x is always positive. TE TE

(39)

This is not the case for the plots of Ycyl . div ., x shown in panel

In terms of energy related to scattering and absorption, the extinction circular dichroism efficiency is obtained from Eq.(35) as,

(b) of Fig. 2. Ycyl . div ., x is positive for all values of kr0,

 , ext

Qext  Q

8  ka

TE TE

suggesting that this component always acts in the direction of wave motion, and induces a repulsive force. As ka increases, TE TE Ycyl decreases, but increases significantly as ka . div ., x

 , ext

Q

approaches kr0.





n  2

n

 ,

TM TE n

.

Panel (c) of Fig. 2 shows the plots for Ycyl . div ., x representing

(40)

the cross-terms or mode conversion occurring between the TM and TE polarizations for the LHCP cylindrical wave field. The

n 0

 ,

plots corresponding to kr0 = 1, 2, 4 and 6 show that Ycyl . div ., x is always negative, which contributes to the generation of a pulling force component toward the source. For kr0 = 10, 13

JQSRT (2020)

Fig. 7. Panels (a)-(f) display the plots for the dimensionless radiation force functions [given by Eqs.(19)-(22)] versus the EM admittance parameter of a Rayleigh PEMC cylinder illuminated by circularly polarized cylindrical diverging waves at ka = 0.1. Panels (g) and (h) display the ORFCD components for plane and cylindrically diverging waves, respectively. For each plot, the dimensionless parameter kr0 has the value indicated in the legend.

14

JQSRT (2020)

Fig. 8. Panels (a)-(f) display the plots for the extinction energy efficiency factors for a Rayleigh PEMC cylinder in circularly polarized cylindrical diverging waves at ka = 0.1. Panel (g) displays the EEECD given by Eq.(40) for cylindrically diverging waves. For the small PEMC cylinder, the results show that the extinction energy efficiency factors are negligibly affected as M and kr0 vary (i.e., all the plots are quasi-superimposed). The solid dashed (black) lines in panels (d) and (f) correspond to the absorption energy efficiency.

15

JQSRT (2020)

Ycyl ,.div ., x is negative for ka < 3, and vanishes around ka = 3. It

results are displayed in panels (a)-(g) of Fig. 3 for all components. In all the cases, and since the PEMC cylinder is lossless, the absorption efficiencies vanish, as required by energy conservation. Panels (a) and (b) of Fig. 3 show that the extinction

becomes positive as ka increases further, and vanishes again around ka = 8.9. Beyond that limit, and as kr0 approaches ka,

Ycyl,.div .,x becomes negative near the source.

TM TM

efficiencies Qext



The plots for the total radiation force function Ycyl . div ., x for 



to that of Ycyl . div ., x shown in panel (a). That is, Ycyl . div ., x is maximal and positive in the Rayleigh regime. It vanishes as the source approaches the PEMC cylinder surface, and becomes negative as the source is located nearby it. For plane 

 ,

Panel (e) of Fig. 2 displays the plots for Ycyl . div ., x corresponding to the interaction of TM and TE polarizations assuming a RHCP cylindrical diverging wave field. The TE TE

,





scattering Qsca and extinction Qext efficiencies versus ka, which are equal and positive, in agreement with energy

behavior of Ycyl . div ., x is somewhat similar to Ycyl . div ., x shown  ,

in panel (b), such that Ycyl . div ., x > 0 for all the values of kr0. The plots for the total radiation force function Y

,

scattering Qsca and extinction Qext efficiencies for LHCP cylindrically diverging waves, which are equal, but negative. This suggests that the interference between TM and TE modes in the interaction of LHCP cylindrically diverging waves with the PEMC cylinder has the tendency to reduce the total scattering. Panel (d) on the other hand, displays the plots for the total

progressive waves, Ycyl . div ., x is always positive.

 cyl . div ., x

+

with previous results of linearly polarized waves (See Eqs.(27) and (28) in [24]). These physical observable are always positive. Panel (c) of Fig. 3 shows the plot for the interference

(d). The behavior of Ycyl . div ., x versus ka is somewhat similar

 ,

TM TM

, amount to Qsca

TM TE TE TE TE TM and Qsca + Qsca , respectively, which agrees Qsca

the LHCP cylindrically diverging waves are displayed in panel TM TM

TE TE

and Qext



conservation, i.e., Qabs  0 . Notice that the PEMC cylinder material is totally reflecting of the EM waves. Only circumferential creeping waves are scattered in the medium of wave propagation [46-48] and no losses can occur in its core material. As the incident field circular polarization changes from

for

the case of RHCP cylindrically diverging waves are displayed 

in panel (f). The variations of Ycyl . div ., x versus ka display also TE TE

some similarities with those of Ycyl . div ., x shown in panel (b),

,

LHCP to RHCP, the interference scattering Qsca

suggesting a repulsive force on the PEMC cylinder using RHCP cylindrically diverging waves for all values of kr0. Comparison of panel (f) with panel (d) shows that the sense of circular polarization has a significant effect on the behavior of the force (i.e., repulsive vs. attractive) experienced by the cylinder. The ORFCD is computed as well, and panel (g) of Fig. 2

and

,

extinction Qext efficiencies become positive, as shown in panel (e) of Fig. 3. Comparison of panel (e) with panel (c) ,

,

 ,

 ,

shows that Qsca = Qext = Qsca = Qext , for all the values of kr0. Panel (f) of Fig. 3 shows the plots for the total scattering

displays the plot of Y p . p .w., x for plane waves. As shown in

  and extinction Qext efficiencies for RHCP cylindrically Qsca

panel (g), Y p . p . w., x is not affected by the variations of kr0

diverging waves versus ka, which are equal and positive, in 

agreement with energy conservation, i.e., Qabs  0 . Subsequently, the EEECD for cylindrically diverging waves

and is independent of it. Furthermore, Y p . p . w., x < 0 for this particular value of the EM cylinder admittance M (= 0.5), and approaches zero as ka increases. A negative ORFCD for plane waves suggests that the radiation force induced by a RHCP plane wave-field is larger than the one induced by LHCP plane waves. Subsequently, the ORFCD for cylindrically diverging waves

is evaluated via Qext , which amounts to Qsca as shown in panel (g) of Fig. 3. Qext < 0 for all the selected kr0 values versus ka, suggesting that extinction by a RHCP cylindrically diverging wave-field is larger than the one induced by LHCP cylindrically diverging waves. Negative EEECD is consistent with numerous experimental studies and findings in analytical chemistry [49], liquid chromatography [50], nucleic acids research [51], biological chemistry, spectroscopy etc. and used as a powerful method in particle characterization. The effect of increasing the EM admittance is investigated, and the radiation force efficiencies and ORFCD components are computed for a PEMC cylinder with M = +1. The corresponding results are displayed in panels (a)-(f) of Fig. 4. The change in the EM admittance alters the radiation force functions, however, some similarities with the results shown in Fig. 2 occur. Concerning the ORFCD, panel (h) of Fig. 4

Ycyl .div ., x is evaluated, and panel (h) of Fig. 2 displays the plots at the selected kr0 values. Some similarity arises between the plots of panel (h) and those displayed in panel (c) for

Ycyl,.div ., x . However, in panel (h), Ycyl .div ., x < 0 for all values of kr0. This also suggests the radiation force induced by a RHCP cylindrically diverging wave-field is larger than the one induced by LHCP cylindrically diverging waves. The scattering, extinction and absorption energy efficiencies are also computed for a lossless PEMC cylinder with an EM admittance M = 0.5 versus ka at different values of kr0, and the 16

JQSRT (2020) shows

that

Ycyl .div ., x changes

sign

for

the

Finally, the effect of varying the admittance of the PEMC cylinder is examined for a Rayleigh (ka = 0.1) and Mie (ka = 5) cylinder, respectively. The radiation force and extinction energy efficiency components are computed in the range –5  M  5 for three values of kr0. For the Rayleigh PEMC cylinder case, kr0 is chosen so that the cylinder is near (kr0 = 0.2), midway (kr0 = 1) and far (kr0 = 10) from the source.

curve

corresponding to kr0 = 10; it is negative for ka < 6, vanishes near ka  6 and turns positive as ka further increases. Moreover, it vanishes again near ka  8.25, and becomes negative as kr0 (= 10) approaches ka. For plane progressive waves (i.e., kr0  ), Ycyl . div ., x < 0 in the range 0 < ka  10.

TM TM

Panel (a) of Fig. 7 displays the plots for Ycyl . div ., x , which

The effect of altering the sign of the admittance is also examined, and numerical computations for the radiation force functions for M = –1 are displayed in panels (a)-(h) of Fig. 5. Visual inspection and comparison with the panels of Fig. 4 show the even functions property, such that TM TM   TE TE      ,     ,          cyl . div ., x

Y

TM TM   TE TE      ,      ,          cyl . div ., x

 ka, M   Y

show symmetry versus the axis M = 0. Notice that M = 0 corresponds to the case of a perfect magnetic cylinder where the effect of the cross-polarized waves vanishes. Near the TM TM

source (kr0 = 0.2), panel (a) shows that Ycyl . div ., x can be negative in the range –0.5  M  0.5. When kr0 (= 1) TM TM

increases, Ycyl . div ., x exhibits negative values in a smaller

 ka,  M  .

TM TM

range. For a large distance (kr0 = 10) Ycyl . div ., x > 0 for all the

On the other

values of M.

hand, the ORFCD for plane waves is an odd function such that

Y p. p. w., x  ka, M   Y p. p. w., x  ka ,  M  .

TE TE

Panel (b) shows the plots for Ycyl . div ., x where symmetry also

Notice,

occurs with respect to the axis M = 0. This is somewhat

however, that Ycyl . div ., x is neither even nor odd such that

TE TE

Ycyl .div ., x  ka , M   Ycyl .div ., x  ka ,  M  .

i.e.,

Next, the variations of the radiation force functions as well as the ORFCD versus kr0 are investigated for a PEMC cylinder having an EM admittance of M = 0.5. Three different dimensionless sizes are considered, namely, ka = 0.1, 1 and 5, respectively. The results are displayed in panels (a)-(h) of Fig. TM TM cyl . div ., x

TE TE

source, Ycyl . div ., x becomes negative for M < –1.8 (or M > 1.8). TE TE

Otherwise, Ycyl . div ., x is always positive for all the values of M as kr0 increases.  ,

The plots for Ycyl . div ., x are displayed in panel (c), where an

the cylinder,

asymmetry arises with respect to the axis M = 0. For positive

vanishes before it turns positive as kr0 increases.

values of M, Ycyl . div ., x < 0, while for negative values of M, the

As the source is moved away from

Y

TE TE TE TE Ycyl Near the . div ., x  kr0 , M   Ycyl . div ., x  kr0 ,  M  .

is negative.

6. Panel (a) shows that near the source Y TM TM cyl . div ., x

TM TM

expected since Ycyl . div ., x (and Ycyl . div ., x ) is an even function;

 ,

opposite occurs. For M = 0, this component related to the interference of TM and TE modes vanishes, which is expected for a material lacking rotary polarization. Panel (d) shows the total radiation force function assuming a LHCP cylindrical diverging wave field. Near the source at kr0

TE TE

Panel (b) shows, however, that Ycyl . div ., x > 0 for all values of kr0. As for the interference term for LHCP cylindrically diverging waves, panel (c) shows that the cases when ka = 0.1  ,

and 1, Ycyl . div ., x < 0, contributing to a negative pulling force  , cyl . div ., x

component. As ka increases (= 5), Y



= 0.2, Ycyl . div ., x displays negative values for 0.25 < M < 3.5.

is negative when



Also, for the case where kr0 = 1, Ycyl . div ., x is negative over a

the source is located nearby the cylinder, vanishes around kr0  6.5, and becomes positive as kr0 further increases. The behavior of the total radiation force function for LHCP cylindrically diverging waves is shown in panel (d) where

narrow range 0.6 < M < 1.4. As the source moves away from 

the cylinder, Ycyl . div ., x is positive.  ,

Ycyl ,.div ., x is only negative near the cylinder. As the incident

Panel (e) displays the plots for Ycyl . div ., x , where for positive

field is changed to a RHCP cylindrically diverging waves, the

values of M, Ycyl . div ., x > 0, while for negative values of M,

 ,

 ,

 ,  , Ycyl . div., x < 0. Similarly to panel (c), Ycyl . div ., x vanishes for M =

radiation force component Ycyl . div ., x is always positive as shown in panel (e). It also contributed to a pushing/repulsive force as shown in panel (f). The ORFCD for plane progressive waves is displayed in panel (g), which clearly shows total independence of the parameter kr0. In all the cases,

0. The plots for the total radiation force function assuming a RHCP cylindrical diverging wave field are shown in panel (f). 

Y p . p . w., x < 0 for the PEMC cylinder with M = 0.5. The

Near the source at kr0 = 0.2, Ycyl . div ., x displays negative values

curves in panel (h) also show that the ORFCD of cylindrically diverging waves is also negative.

Ycyl .div ., x is negative over the range –1.4 < M < –0.6, while

for –3.5 < M < –0.25. Also, for the case where kr0 = 1,

Ycyl .div ., x > 0 for kr0 = 10. 17

JQSRT (2020)

Fig. 9. The same as in Fig. 7, but ka = 5.

18

JQSRT (2020)

Fig. 10. The same as in Fig. 8, but ka = 5.

19

JQSRT (2020) both LHCP and RHCP waves, which could be useful in optical manipulation and in studying the mechanical effects of EM waves on particles exhibiting rotary polarization. In this analysis, a rigorous analytical formalism is developed and examplified for a lossless PEMC circular cylinder using the modal expansion method in partial-wave series in cylindrical coordinates. Mathematical expressions for the EM radiation forces (per-length) as well as the scattering, extinction and absorption energy efficiencies are derived without any approximations. The incident illuminating field is either composed of LHCP or RHCP cylindrical diverging waves. Numerical computations are performed with particular emphases on the size parameter of the cylinder, its distance from the source and EM admittance parameter. Several conditions are predicted where the PEMC cylinder located nearby the source can experience a pulling negative force in opposite direction of wave propagation. Should some conditions related to the size parameter, source distance and admittance parameter be met, the PEMC cylinder becomes irresponsive to the linear transfer of momentum, and experiences zero force. In addition, the possibility of EM cloaking is also predicted for a Rayleigh PEMC cylinder illuminated by LHCP or RHCP waves. Furthermore, both the ORFCD and EEECD alternate between negative or positive values depending on the EM admittance. This effect may be used in spectroscopy methods in particle characterization and classification. The results show that the numerical predictions are in total agreement with the law of energy conservation. It is important to note here that the radiation force expressions developed here are not restricted to the case of a PEMC cylinder, but can be used for any cylindrical material exhibiting polarization rotation such as a chiral, topological insulator or plasma cylinder to name a few cases. Concerning cylindrical particles with geometrical crosssections deviating from the circular one, recent investigations have addressed part of this challenge for various cases of illuminations with linearly polarized fields [52-55]. It would be interesting to further investigate the ORFCD and EEECD methods for the characterization of elliptical cylinders with smooth and corrugated surfaces, and the influence of either LHCP or RHCP waves on the radiation forces for a PEMC cylinder. Moreover, the analysis to consider the effects of boundaries/corners on the ORFCD and EEECD for a PEMC cylinder would be of interest, and the scope of the previous works on homogeneous particles [56, 57] could be extended for this purpose. Also, multiple scattering effects [52, 58, 59], which are significant in the presence of two or more cylinders, affect the ORFCD and EEECD spectroscopy methods. Further analyses are warranted to carefully examine potentially novel effects in radiation force and EM cloaking arising from those emergent phenomena.

The ORFCD for plane progressive waves is displayed in panel (g), which exhibits positive or negative values when the admittance is negative or positive, respectively. Similarly, the ORFCD for cylindrical diverging waves displayed in panel (h) reveals positive or negative variations versus negative or positive admittance, respectively, which is the largest near the source. The extinction energy efficiency factors are also computed, and the results are displayed in panels (a)-(g) of Fig. 8 for the Rayleigh (i.e., small) PEMC cylinder at ka = 0.1. Notice that in all the plots, the extinction factors are minimally affected by the change of the source-cylinder dimensionless distance (i.e., all the plots are superimposed for the three selected values of TM TM

TE TE

kr0). Panels (a) and (b) show that Qext and Qext are symmetric with respect to the axis M = 0 and are always positive. The interference extinction factor displayed in panel (c) exhibits positive and negative values for negative and positive admittance, respectively, while the total extinction assuming a LHCP cylindrical diverging waves is always positive as required by energy conservation and shown in panel (d). The solid dashed line in panel (d) [also in panel (f)] corresponds to the absorption efficiency, which is zero as expected. An interesting observation is noted for M = +1 for 

all values of kr0, where Qext  0. This effect suggests the feasibility of EM cloaking of a Rayleigh PEMC cylinder in LHCP waves where extinction is nearly zero. Panel (e) for the interference extinction factor for RHCP waves varies between negative and positive values as M varies similarly, and vanishes for M = 0. Panel (f) shows the total extinction energy efficiency factor for RHCP waves, where also the possibility of achieving EM cloaking is predicted for M = –1. The EEECD for the Rayleigh PEMC cylinder is displayed in panel (g), where positive and negative values are computed while M varies, respectively, from negative to positive numbers. Thus, it can be concluded that the EEECD effect is highly sensitive to the EM admittance parameter, which can be used as a potential means in particle characterization of particles allowing rotary polarization. Additional computations versus admittance are performed for a larger Mie PEMC cylinder with ka = 5 at three different values of kr0 (= 5.1, 6 and 10). The corresponding results for the radiation force functions and extinction efficiency factors are displayed in the panels of Figs. 9 and 10, respectively. Although the plots in the panels of Fig. 9 display somewhat similar behaviors with the Rayleigh case shown in Fig. 7, the amplitudes are smaller. On the other hand, the panels in Fig. 10 for the Mie PEMC cylinder exhibit significantly different characteristics, quite distinct from the Rayleigh case presented in Fig. 8. Particularly, panels (d) and (f) in Fig. 10 do not predict any cloaking effect, which only occurs for the small PEMC cylinder. Moreover, the distance from the source affects the extinction efficiency factors distinctively.

References

4. Conclusion and some perspectives

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This work introduces a method of spectroscopy, termed here optical radiation force circular dichroism (ORFCD). This technique is based on subtracting the radiation forces of 20

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Author Statement

FG Mitri has conceptualized the methodology, its validation, data curation, and writing of the paper in its entirety.

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