111∗ Corresponding author<EMAIL_ADDRESS>
# Intrinsic angular momentum, spin and helicity of higher-order Poincaré modes

M. Babiker1,∗, K. Koksal2, V. E. Lembessis3 and J. Yuan1 1School of Physics
Engineering and Technology, University of York, YO10 5DD, UK  2Physics
Department, Bitlis Eren University, Bitlis, Turkey  3Quantum Technology Group,
Department of Physics and Astronomy, College of Science, King Saud University,
Riyadh 11451, Saudi Arabia

###### Abstract

The availability of coherent sources of higher order Poincaré optical beams
have opened up new opportunities for applications such as in the optical
trapping of atoms and small particles, the manipulation of chirally-sensitive
systems and in improved encoding schemes for broad-bandwidth communications.
Here we determine the intrinsic properties of Poincaré Laguerre-Gaussian (LG)
modes which have so far neither been evaluated, nor their significance
highlighted. The theoretical framework we adopt here is both novel and
essential because it emphasises the crucial role played by the normally
ignored axial components of the twisted light fields of these modes. We show
that the inclusion of the axial field components enables the intrinsic
properties of the Poincaré modes, notably their angular momentum, both spin
and orbital as well as their helicity and chirality, to be determined. We
predict significant enhancements of the intrinsic properties of these modes
when compared with those due to the zero order LG modes. In particular, we
show that higher order LG Poincaré modes exhibit super-chirality and,
significantly so, even in the case of the first order

##  I 1\. Introduction

Recent advances in the field of twisted light have recognised the importance
and the prospects for useful applications of the so-called higher order
Poincaré vortex modes Vaziri  _et al._ (2002); Milione  _et al._ (2011);
Galvez  _et al._ (2021); Maurer  _et al._ (2007); Zhan (2009); Liu  _et al._
(2014); Fickler  _et al._ (2014); Krenn  _et al._ (2014); Ren  _et al._
(2015); Naidoo  _et al._ (2016); Liu  _et al._ (2017); Volpe and Petrov
(2004); Holmes and Galvez (2019); Chen  _et al._ (2020). For a given order
specified by the integer $m\geq 0$, the optical polarisation
${\mbox{\boldmath$\hat{\epsilon}$}}_{m}$ arises as a non-separable
superposition of the circular polarisation states
$({\mbox{\boldmath$\hat{x}$}}\pm i{\mbox{\boldmath$\hat{y}$}})/\sqrt{2}$ with
spatial phase $e^{\pm im\phi}$. Every state of polarisation is identified
uniquely by a point $(\Theta_{P},\Phi_{P})$ on the surface of a unit higher
order Poincaré sphere (HOPS), as shown and explained in the caption of Fig.1.

Note that the positive order parameter $m$ includes the lowest orders, namely
$m=0$ and $m=1$, to be referred to, respectively, as the basic 0th-order and
first order set of modes. Each is distinguished uniquely by its own order
Poincaré sphere. The Poincaré sphere for $m=0$ is scalar and incorporates all
elliptically-polarised (including circularly- and linearly- polarised)
Gaussian modes. The first order $m=1$ has a different Poincaré sphere and
includes, in addition, the radially- and azimuthally-polarised optical vortex
modes Volpe and Petrov (2004).

Although several experimental reports have already confirmed the controlled
generation of the higher order vector modes Naidoo  _et al._ (2016); Liu  _et
al._ (2017); Chen  _et al._ (2020), the optical properties for arbitrary
higher order $m\geq 0$ are, as far as we know, not yet been evaluated. Both
the spin angular momentum (SAM) and the optical angular momentum (AM), as well
as the orbital angular momentum (OAM) are yet to be evaluated for a general
order $m$ Laguerre-Gaussian higher order modes. It is not clear what spatial
density variations the vector components of the SAM, OAM and AM of these modes
have and what $m$-dependence each has.

It is also clearly important to find out whether and how the higher order
modes can lead to enhanced optical properties and enable new applications such
as higher order encoding protocols for significantly increased bandwidth in
quantum communication Al-Amri  _et al._ (2021) and whether their angular
momentum, spin and chirality are greatly enhanced. Evidently, enhanced optical
chirality is highly desirable for chiroptical processes Tang and Cohen (2010).
We show that higher order Poincaré modes exhibit super-chirality and,
significantly so, even for the first order for which $m=1$. For recent
accounts on the optical interaction with chiral matter the reader is referred
to Mun  _et al._ (2020).

The theoretical framework we adopt here aims first at evaluating the energy
flux and the linear momentum density and proceeds to determine the cycle-
averaged spin angular momentum density, the total angular momentum density,
the orbital angular momentum density as well as the helicity and the chirality
densities. For each of these main intrinsic properties we determine the
spatial density distributions followed by the evaluation of its spatial
integral over the focal plane at $z=0$. The evaluations are carried out using
the electromagnetic fields associated with the most general paraxial mode of
arbitrary order $m$($\geq 0$). An important feature of this paper is that
although our main focus is on Laguerre-Gaussian modes our treatment is
applicable to any type of twisted light and includes all the possible
scenarios of optical polarisation of higher order Poincaré modes.

This paper is organised as follows. In section 2 we begin with the vector
potential ${\bf A}$ for a general paraxial optical vortex endowed with the
higher order polarisation state ${\mbox{\boldmath$\hat{\epsilon}$}}_{m}$ which
is defined in terms of the higher order Poincaré sphere. We then describe the
steps leading to the electric and magnetic fields of the higher order modes.
In section 3 we define the cycle-averaged intrinsic properties and proceed to
evaluate their respective densities for the higher order modes for each of the
stated properties, namely the SAM, the AM and the OAM, as well as the helicity
and the chirality. Each evaluated density is then followed by the evaluation
of the integrated (total) property per unit length. The $m$-dependence of the
helicity density has been elucidated in Babiker  _et al._ (2024), so the
emphasis in this paper is the demonstration ofof the involvement of both the
higher order parameter $m$ as well as the radial number $p$, as for Laguerre-
Gaussian modes LG${}^{m}_{p}$. Section 4 summarises the results and comments
on the significance of these results and Appendices A and B discuss the
normalisation factor ${\cal A}_{0}$ in terms of the Power ${\cal P}_{T}$ and
we consider the four definite integrals needed in the body of the paper.

##  II 2\. Fields of higher order Poincaré modes

The paraxial electromagnetic fields associated with an optical vortex in a
higher order state of polarisation are derivable from a vector potential in
cylindrical coordinates ${\bf r}=(\rho,\phi,z)$ in the form

${\bf A}={\mbox{\boldmath$\hat{\epsilon}$}}_{m,p}{\tilde{\cal
F}}_{m,p}(\rho,z)e^{ik_{z}z}$ (1)

where ${\mbox{\boldmath$\hat{\epsilon}$}}_{m,p}$ is the order $m$ polarisation
and its variations cover every point on the surface of the order order $m\geq
0$ unit Poincaré sphere. This is as follows

${\mbox{\boldmath$\hat{\epsilon}$}}_{m,p}=e^{im\phi}\frac{({\mbox{\boldmath$\hat{x}$}}-i{\mbox{\boldmath$\hat{y}$}})}{\sqrt{2}}\cos\left(\frac{\Theta_{P}}{2}\right)e^{-i\Phi_{P}/2}+e^{-im\phi}\frac{({\mbox{\boldmath$\hat{x}$}}+i{\mbox{\boldmath$\hat{y}$}})}{\sqrt{2}}\sin{\left(\frac{\Theta_{P}}{2}\right)}e^{i\Phi_{P}/2}$
(2)

where $\Theta_{P}$ and $\Phi_{P}$ are Poincare sphere angles as defined in
Figure 1. The vector ${\mbox{\boldmath$\hat{\epsilon}$}}_{m}$ is the most
general polarisation state vector, and written using our convention as
depicted in Fig. 1 so it differs slightly from the form defined by Milione et
al Milione  _et al._ (2011). The validity of the polarisation states for the
Poincaré modes, has already been confirmed experimentally Liu  _et al._
(2014); Naidoo  _et al._ (2016); Chen  _et al._ (2020).

In Eq.(1) the polarisation is seen multiplying the paraxial vortex mode
function ${\tilde{\cal F}}_{m,p}(\rho,z)e^{ik_{z}z}$ which has no
$\phi$-dependence and in which $k_{z}$ is the wavevector for the light
travelling along the $+z$ axis and ${\cal F}_{m,p}$ specifying only the
amplitude variations in terms of the coordinates $(\rho,z)$. The mode function
is labelled by the its integer order $m>0$, (which is also the winding number)
and $p\geq 0$ (which is the radial number), as for a Laguerre-Gaussian (LG)
optical vortex modes.

Figure 1: 0th order, (a), and 1st-order, (b), Pioncaré Sphere (PS)
representation of the polarisation state in which optical polarisation is
coupled with vortex phase, characterized by a unit sphere with respect to the
corresponding Stokes-parameter(-like) Cartesian coordinates
($S^{0}_{1},S^{0}_{2}$, $S^{0}_{3}$) and ($S^{1}_{1},S^{1}_{2}$, $S^{1}_{3}$)
respectively. It is seen that the 0th order PS is equivalent to the
conventional PS where $|H>$ and $|V>$ are commonly used to denote the
vertically and horizontally linearly polarized light, $|A>$ and $|D>$ for $\pm
45^{o}$ tilted linearly polarized light, $|R>$ and $|L>$ for right-hand and
left-hand circularly-polarized light, respectively. The 1st order PS figure is
related to the corresponding figure by Milione et al Milione  _et al._ (2011)
with slightly different conventions for $S^{1}_{1}$ and $S^{1}_{2}$. Six sets
of special vector modes are drawn in different colours next to each sphere for
illustration. Their positions on the Poincaré sphere are indicated by dots of
the same colour.

In order to simplify the evaluations we now express the polarisation vector
${\mbox{\boldmath$\hat{\epsilon}$}}_{m}$, Eq.(3), in the following form

${\mbox{\boldmath$\hat{\epsilon}$}}_{m,p}=e^{im\phi}({\mbox{\boldmath$\hat{x}$}}-i{\mbox{\boldmath$\hat{y}$}}){\cal
U}_{P}+e^{-im\phi}({\mbox{\boldmath$\hat{x}$}}+i{\mbox{\boldmath$\hat{y}$}}){\cal
V}_{P}$ (3)

where ${\cal U}_{P}$ and ${\cal V}_{P}$ are the complex Poincare’ angular
functions (i.e. of $\Theta_{P},\Phi_{P}$) given by

${\cal
U}_{P}=\frac{1}{\sqrt{2}}\cos{\left(\frac{\Theta_{P}}{2}\right)}e^{i\Phi_{P}/2};\;\;\;\;{\cal
V}_{P}=\frac{1}{\sqrt{2}}\sin{\left(\frac{\Theta_{P}}{2}\right)}e^{-i\Phi_{P}/2}$
(4)

The Poincare angular functions satisfy the following identities

$\displaystyle({\cal U}_{P}^{*}{\cal V}_{P}-{\cal V}_{P}^{*}{\cal U}_{P})$
$\displaystyle=$ $\displaystyle-\frac{i}{2}\sin(\Theta_{P})\sin(\Phi_{P})$
$\displaystyle(|{\cal U}_{P}|^{2}-|{\cal V}_{P}|^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\cos(\Theta_{P})$ $\displaystyle(|{\cal
U}_{P}|^{2}+|{\cal V}_{P}|^{2})$ $\displaystyle=$ $\displaystyle\frac{1}{2}$
(5)

We seek to develop the analysis for a general ${\tilde{\cal F}}$, which could
be appropriate for any optical vortex type, for example Laguerre-Gaussian,
Bessel, Bessel-Gaussian,..etc., and then consider the evaluations for the
Laguerre-Gaussian case. We begin by writing the vector potential in Eq.(1) in
the following form

${\bf
A}=\left\\{({\mbox{\boldmath$\hat{x}$}}-i{\mbox{\boldmath$\hat{y}$}})e^{im\phi}{\cal
U}_{P}+({\mbox{\boldmath$\hat{x}$}}+i{\mbox{\boldmath$\hat{y}$}})e^{-im\phi}{\cal
V}_{P}\right\\}{\tilde{\cal F}}_{m,p}(\rho,z)e^{ik_{z}z}$ (6)

This form of the generally-polarised mode of order $m$ suggests that it is a
superposition of two vortex modes, one with winding number $m$ and right-
handed circular polarisation, weighted by the Poincaré function ${\cal U}_{P}$
and the second has winding number $-m$ and left-handed circular polarisation,
weighted by the Poincaré function ${\cal V}_{P}$.

One of the main requirements to be satisfied by free-space paraxial optical
fields is that the electric field must be derivable from the magnetic field
using the Maxwell curl equation and that the electric field must produce the
same magnetic field via the second Maxwell curl equation. For convenience, we
can do so by writing the vector potential Eq.(6) as the sum of two parts as
follows:

${\bf A}={\bf A_{1}}+{\bf A_{2}}$ (7)

where

${\bf A_{1}}=({\mbox{\boldmath$\hat{x}$}}-i{\mbox{\boldmath$\hat{y}$}}){\cal
F}^{(1)}_{m,p}({\bf r})e^{ik_{z}z};\;\;\;\;{\bf
A_{2}}=({\mbox{\boldmath$\hat{x}$}}+i{\mbox{\boldmath$\hat{y}$}}){\cal
F}^{(2)}_{m,p}({\bf r})e^{ik_{z}z}$ (8) ${\cal F}^{(1)}_{m,p}({\bf r})={\cal
U}_{P}e^{im\phi}{\tilde{\cal F}}_{m,p}(\rho,z);\;\;\;{\cal F}^{(2)}_{m,p}({\bf
r})={\cal V}_{P}e^{-im\phi}{\tilde{\cal F}}_{m,p}(\rho,z)$ (9)

Note that so far we have regarded ${\tilde{\cal F}}$ as a general optical
vortex mode amplitude function and here we develop the formalism without
specifying this function. Once the results are arrived at, we shall consider
the special case of ${\tilde{\cal F}}$ appropriate for Laguerre-Gaussian
modes.

The magnetic and electric fields of our generally-polarised mode are written
as the sums of two terms ${\bf B}={\bf B}_{1}+{\bf B}_{2}$ and ${\bf E}={\bf
E}_{1}+{\bf E}_{2}$ where ${\bf B}_{i}={\mbox{\boldmath$\nabla$}}\times{\bf
A}_{i};\;\;\;i=1,2$. The sequence of steps leading to the required expressions
for the fields involve dealing with the two parts of the magnetic field first
and from those use Maxwell’s curl B equation to derive the corresponding
electric field parts. We have

$\displaystyle{\bf B}_{1}$ $\displaystyle=$ $\displaystyle
ik_{z}({\mbox{\boldmath$\hat{y}$}}+i{\mbox{\boldmath$\hat{x}$}}){\cal
F}^{(1)}e^{ik_{z}z}-{\mbox{\boldmath$\hat{z}$}}\left(i\frac{\partial{\cal
F}^{(1)}}{\partial x}+\frac{\partial{\cal F}^{(1)}}{\partial
y}\right)e^{ik_{z}z}$ $\displaystyle{\bf E}_{1}$ $\displaystyle=$
$\displaystyle
ick_{z}({\mbox{\boldmath$\hat{x}$}}-i{\mbox{\boldmath$\hat{y}$}}){\cal
F}^{(1)}e^{ik_{z}z}-{\mbox{\boldmath$\hat{z}$}}c\left\\{\frac{\partial{\cal
F}^{(1)}}{\partial x}-i\frac{\partial{\cal F}^{(1)}}{\partial
y}\right\\}e^{ik_{z}z}$ $\displaystyle{\bf B}_{2}$ $\displaystyle=$
$\displaystyle
ik_{z}({\mbox{\boldmath$\hat{y}$}}-i{\mbox{\boldmath$\hat{x}$}}){\cal
F}^{(2)}e^{ik_{z}z}+{\mbox{\boldmath$\hat{z}$}}\left(i\frac{\partial{\cal
F}^{(2)}}{\partial x}-\frac{\partial{\cal F}^{(2)}}{\partial
y}\right)e^{ik_{z}z}$ $\displaystyle{\bf E}_{2}$ $\displaystyle=$
$\displaystyle
ick_{z}({\mbox{\boldmath$\hat{x}$}}+i{\mbox{\boldmath$\hat{y}$}}){\cal
F}^{(2)}e^{ik_{z}z}-{\mbox{\boldmath$\hat{z}$}}c\left\\{\frac{\partial{\cal
F}^{(2)}}{\partial x}+i\frac{\partial{\cal F}^{(2)}}{\partial
y}\right\\}e^{ik_{z}z}$

where we have dropped the subscript labels in ${\cal F}^{(1),(2)}$ and in
${\tilde{\cal F}}$ for ease of notation and these can be restored when the
need arises. The fields we have derived in the set of equations
(LABEL:emfields) form the basis for the derivation of the optical properties
of the order $m$ vector vortex mode. Note that up to now we have not specified
the type of vortex mode and we continue to deal with the general form
involving the vector potential amplitude function ${\tilde{\cal
F}}_{m,p}(\rho,z)$.

##  III 3\. Cycle-averaged intrinsic properties

Our aim is to evaluate all the major cycle-averaged properties of the higher
order Poincaré mode. These include the optical spin angular momentum (SAM)
density ${\bar{\bf s}}$ the angular momentum (AM) density $\bar{j}$ from which
we obtain the orbital angular momentum (OAM) density and finally, the helicity
and the chirality of the mode. The cycle-averaged AM density requires
evaluation of the linear momentum density and we also need to evaluate the
power flux for normalisation. The relevant properties are defined as follows
Allen  _et al._ (1999)

$\bar{s}$ $\displaystyle=$
$\displaystyle\frac{1}{4\omega}\Im\left\\{[\epsilon_{0}{\bf E}^{*}\times{\bf
E}]+\frac{1}{\mu_{0}}[{\bf B}^{*}\times{\bf B}]\right\\};$ (11)
$\displaystyle=$
$\displaystyle{\mbox{\boldmath$\bar{s}$}}_{E}+{\mbox{\boldmath$\bar{s}$}}_{B}\;\;\;\;\;\;{\rm{(SAM\;density)}}$
${\mbox{\boldmath$\bar{j}$}}={\bf
r}\times{\mbox{\boldmath$\bar{\pi}$}};\;\;\;\;({\rm{AM\;density}})$ (12)
${\mbox{\boldmath$\bar{\ell}$}}={\mbox{\boldmath$\bar{j}$}}-{\mbox{\boldmath$\bar{s}$}};\;\;\;\;\;({\rm{OAM\;density}})$
(13)

$\displaystyle{\bar{\eta}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{c}{\omega^{2}}{\bar{\chi}}=-\frac{\epsilon_{0}c}{2\omega}\Im{[{\bf
E}^{*}\cdot{\bf B}]}$ (14)
$\displaystyle({\rm{Helicity/\;Chirality\;density}}),$

where in the above ${\bar{\mbox{\boldmath$\pi$}}}=\frac{1}{c^{2}}{\bar{\bf
w}}$ is the linear momentum density with ${\bar{\bf
w}}=\frac{1}{2\mu_{0}}\Re[{\bf E}^{*}\times{\bf B}]$ the energy density. The
symbols ${\Re}[...]$ and $\Im[...]$ stand for real and imaginary parts of […]
and the superscript * in ${\bf E}^{*}$ stands for the complex conjugate of
${\bf E}$. Note that in free space the chirality density ${\bar{\chi}}$ is
proportional to the helicity density ${\bar{\eta}}({\bf r})$, so we only need
to deal with the helicity from which the chirality follows.

As stated above we consider the evaluations of the above densities and deal
with them in turn, including their vector components, all evaluated
specifically in relation to the higher order Poincaré modes. After this the
next tasks will involve the evaluation of the total intrinsic properties per
unit length for SAM and AM and from these two we deduce the OAM. Finally we
evaluate the helicity and chirality per unit length. Each property is
evaluated as the space integral of the corresponding density over the x-y
plane.

The evaluation now requires as a first step expressions for the $x$\- and
$y$-derivatives of ${\cal F}^{(1)}$ in polar coordinates. Note that ${\cal
F}^{(1)}$ is distinguished by the phase factor $\exp{(+im\phi)}$ and ${\cal
F}^{(2)}$ is distinguished by the phase factor $\exp{(-im\phi)}$. We have for
${\cal F}^{(1)}$

$\frac{\partial{\cal F}^{(1)}}{\partial x}={\cal
U}_{P}\left\\{\cos\phi{\tilde{\cal
F}^{\prime}}-i\frac{m}{\rho}\sin\phi{\tilde{\cal F}}\right\\}e^{im\phi}$ (15)
$\frac{\partial{\cal F}^{(1)}}{\partial y}={\cal
U}_{P}\left\\{\sin\phi{\tilde{\cal
F}^{\prime}}+i\frac{m}{\rho}\cos\phi{\tilde{\cal F}}\right\\}e^{im\phi}$ (16)
$\frac{\partial{\cal F}^{(2)}}{\partial x}={\cal
V}_{P}\left\\{\cos\phi{\tilde{\cal
F}^{\prime}}+i\frac{m}{\rho}\sin\phi{\tilde{\cal F}}\right\\}e^{-im\phi}$ (17)

and

$\frac{\partial{\cal F}^{(2)}}{\partial y}={\cal
V}_{P}\left\\{\sin\phi{\tilde{\cal
F}^{\prime}}-i\frac{m}{\rho}\cos\phi{\tilde{\cal F}}\right\\}e^{-im\phi}$ (18)

###  III.1 Paraxial Laguerre-Gaussian forms

Ultimately we will apply the formalism to the case of higher order Laguerre-
Gaussian Poincaré modes. We thus require the amplitude function of these for
winding number $m$ and radial number $p$ which is

${\tilde{\cal F}}_{m,p}(\rho,z)={\cal
A}_{0}\frac{C_{|m|,p}}{\sqrt{1+z^{2}/z_{R}^{2}}}\left(\frac{\rho\sqrt{2}}{w_{0}\sqrt{1+z^{2}/z_{R}^{2}}}\right)^{|m|}\exp\left[\frac{-\rho^{2}}{w_{0}^{2}(1+z^{2}/z_{R}^{2})}\right]L_{p}^{|m|}\left\\{\frac{2\rho^{2}}{w_{0}^{2}(1+z^{2}/z_{R}^{2})}\right\\}e^{i\xi(\rho,z)},$
(19)

where ${\cal A}_{0}$ is a normalisation factor, to be determined in terms of
the power ${\cal P}_{T}$. The phase function $\xi(\rho,z)$ includes the Gouy
and the curvature phases

$\xi(\rho,z)=-(2p+|m|+1)\arctan\left(\frac{z}{z_{R}}\right)+\frac{kz\rho^{2}}{2(z^{2}+z_{R}^{2})}.$
(20)

Here $w_{0}$ is the beam waist, $z_{R}=w_{0}^{2}k/2$ is the Rayleigh range,
$C_{|m|,p}=\sqrt{p!/(p+|m|)!}$ and $L_{p}^{|m|}$ the associated Laguerre
polynomial.

Note that the amplitude function ${\tilde{\cal F}}$ for LG modes only depends
on the absolute value of $m$, so that we can now interpret the $m$-th order
Poincaré modes, defined in in (7, 8, 9), as LGm,p and LG-m,p modes. This fact
is useful experimentally as well as in aiding the interpretation of our
results.

Our main concern will be on evaluating the variations of each intrinsic
property on the focal plane at $z=0$ so that the vector potential amplitude
function reduces to

${\tilde{\cal F}}_{m,p}(\rho)={\cal
A}_{0}\sqrt{\frac{p!}{(p+|m|)!}}e^{-\frac{\rho^{2}}{w_{0}^{2}}}\left(\frac{\sqrt{2}\rho}{w_{0}}\right)^{|m|}L^{|m|}_{p}\left(\frac{2\rho^{2}}{w_{0}^{2}}\right),$
(21)

where now ${\tilde{\cal F}}_{m,p}$ depends only on the radial coordinate
$\rho$. In Appendix A we evaluate the normalisation constant ${\cal A}_{0}$
and express it in terms of the applied power ${\cal P}_{T}$.

###  III.2 The SAM density

The SAM density Eq.(11) receives contributions from both the electric field
${\bar{\bf s}}_{E}$ and the magnetic field ${\bar{\bf s}}_{B}$. We begin by
the evaluation of the first part ${\bar{\bf s}}_{E}$ of the SAM density which
satisfies

$\frac{4\omega}{\epsilon_{0}}{\bar{\bf s}}_{E}=\Im[{\bf E}^{*}\times{\bf E}].$
(22)

Since ${\bf E}={\bf E}_{1}+{\bf E}_{2}$, the vector cross product contains
four products, two of which are direct terms and two mixed terms

${\bf E}^{*}\times{\bf E}=\left\\{({\bf E}_{1}^{*}\times{\bf E}_{1}+{\bf
E}_{2}^{*}\times{\bf E}_{2})+({\bf E}_{1}^{*}\times{\bf E}_{2}+{\bf
E}_{2}^{*}\times{\bf E}_{1})\right\\},$ (23)

where the first set of terms are direct and the second set are the mixed
terms. A similar treatment is followed for the second part of the SAM density.
We have

$4\omega\mu_{0}{\bar{\bf s}}_{B}=\Im[{\bf B}^{*}\times{\bf B}].$ (24)

Here also the vector product consists of four products two of which are direct
and two are mixed

${\bf B}^{*}\times{\bf B}=({\bf B}_{1}^{*}\times{\bf B}_{1}+{\bf
B}_{2}^{*}\times{\bf B}_{2})+({\bf B}_{1}^{*}\times{\bf B}_{2}+{\bf
B}_{2}^{*}\times{\bf B}_{1}).$ (25)

The evaluation of the SAM density is straightforward, but rather lengthy. Each
of the eight vector products consists of three vector components which we
evaluated separately by direct substitutions of the electric and magnetic
fields given in Eqs.(LABEL:emfields).

Extensive careful evaluations lead to the following results. We have found
that the direct E terms are equal to the direct B terms

$\left[{\bf E}_{1,2}^{*}\times{\bf E}_{1,2}\right]=c^{2}\left[{\bf
B}_{1,2}^{*}\times{\bf B}_{1,2}\right].$ (26)

We have also found that all components of the mixed E terms are equal and
opposite in sign to those of the B term. We have

$\left[{\bf E}_{1}^{*}\times{\bf E}_{2}\right]=-c^{2}\left[{\bf
B}_{1}^{*}\times{\bf B}_{2}\right].$ (27)

Similarly we have found

$\left[{\bf E}_{2}^{*}\times{\bf E}_{1}\right]=-c^{2}\left[{\bf
B}_{2}^{*}\times{\bf B}_{1}\right].$ (28)

Thus the direct terms add while mixed ones cancel. Making use of the above
relations we get the simple result

$[{\bf E}^{*}\times{\bf E}]+c^{2}[{\bf B}^{*}\times{\bf B}]=2\left\\{[{\bf
E}_{1}^{*}\times{\bf E}_{1}]+[{\bf E}_{2}^{*}\times{\bf E}_{2}]\right\\}.$
(29)

This simplifies the analysis.

###  III.3 Transverse SAM density components

The x-component of the spin density is

${\bar{s}}_{x}=\frac{\epsilon_{0}}{2\omega}\Im\left\\{[{\bf
E}_{1}^{*}\times{\bf E}_{1}]_{x}+[{\bf E}_{2}^{*}\times{\bf
E}_{2}]_{x}\right\\}.$ (30)

Direct evaluations using the expressions in Eq.(LABEL:emfields) for ${\bf
E}_{1}$ and ${\bf E}_{2}$ lead us to the result

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bar{s}}_{x}=\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\left\\{m\frac{{\tilde{\cal
F}}^{2}}{\rho}+{\tilde{\cal F}^{\prime}}{\tilde{\cal F}}\right\\}\sin\phi},$
(31)

where we have made use of the identities in Eqs.(5). Similarly on evaluating
the y-component and obtain

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bar{s}}_{y}=-\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\left\\{m\frac{{\tilde{\cal
F}}^{2}}{\rho}+{\tilde{\cal F}^{\prime}}{\tilde{\cal F}}\right\\}\cos\phi,}$
(32)

These combine to give the transverse SAM vector component
${\mbox{\boldmath$\bar{s}$}}_{\bot}$ defined as follows

${\mbox{\boldmath$\bar{s}$}}_{\bot}={\bar{s}}_{x}{\mbox{\boldmath$\hat{x}$}}+{\bar{s}}_{y}{\mbox{\boldmath$\hat{y}$}}.$
(33)

This result can be written in terms of the azimuthal unit vector $\hat{\phi}$
as follows

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mbox{\boldmath$\bar{s}$}}_{\bot}=-\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\left\\{m\frac{{\tilde{\cal
F}}^{2}}{\rho}+{\tilde{\cal F}^{\prime}}{\tilde{\cal
F}}\right\\}{\mbox{\boldmath$\hat{\phi}$}}}$ (34)

The emergence of the transverse spin is a direct consequence of the
longitudinal field components in Eqs. (LABEL:emfields) whose strength involves
both the $\phi$ and $\rho$ gradients in the transverse plane Aiello  _et al._
(2015). The $\phi-$gradient accounts for the $m-$dependence in the first term,
while the radial gradient is manifest in the second term ${\hat{\cal
F}^{\prime}}{\hat{\cal F}}$.

###  III.4 The longitudinal component of the SAM density

The longitudinal spin density component is given by

${\bar{s}}_{z}=-\frac{\epsilon_{0}}{2\omega}\Im\left\\{[{\bf
E}_{1}^{*}\times{\bf E}_{1}]_{z}+[{\bf E}_{2}^{*}\times{\bf
E}_{2}]_{z}\right\\}.$ (35)

Similar evaluations lead to the result

$\displaystyle{\bar{s}}_{z}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}[|{\cal U}_{P}^{2}|-|{\cal
V}_{P}^{2}|]{\tilde{\cal F}}^{2}$ (36) $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\cos(\Theta_{P}){\tilde{\cal
F}}^{2}.$

The lack of any $m-$dependence in ${\bar{s}}_{z}$ in Eq. (36) is easy to see
since ${\bar{s}}_{z}$ depends only on contributions arising from the
transverse field components.

This completes the evaluations of the general form of the transverse and the
longitudinal components of the SAM density, which are applicable to any
optical vortex amplitude function ${\tilde{\cal F}}$. However, we shall
proceed to consider the specific case in which ${\tilde{\cal F}}$ corresponds
to a Laguerre-Gaussian amplitude function. The variations of the SAM density
components with the radial coordinates are displayed in Fig.(2).

Figure 2: The variations of the SAM density components (arbitrary units) with
the radial coordinate $\rho$ (in units of $w_{0})$. Here the blue curve
represents the magnitude of the transverse component (Eq. 34) and the yellow
curve represents the longitudinal $z$ component (Eq. 36). For illustration we
have chosen to consider the order $m=2$ and the point on the north pole
$\Theta_{p}=0,\Phi_{p}=0$ of the order m=2 Poincaré sphere.

###  III.5 Integrated SAM density

We have seen that the transverse SAM density components are proportional to
sine and cosine of the azimuthal angle $\phi$ and so on spatial integration
they vanish

${\bar{S}}_{x}=0;\;\;\;\;{\bar{S}}_{y}=0.$ (37)

Only the z-component survives on integration, so we have, on substituting from
Eq.(35)

${\bar{S}}_{z}=\int_{0}^{2\pi}d\phi\int_{0}^{\infty}{\bar{s}}_{z}\rho\;d\rho=\frac{\pi\epsilon_{0}}{\omega}\cos(\Theta_{P}){\cal
I}_{P},$ (38)

and on substituting for ${\cal I}_{P}$ in terms of ${\cal P}_{T}$ using
Eq.(74) we have

${\bar{S}}_{z}=\frac{c^{2}k_{z}^{2}\pi\epsilon_{0}}{\omega}\cos(\Theta_{P})\frac{\mu_{0}{\cal
P}_{T}}{\pi ck_{z}^{2}},$ (39)

which finally leads us to the total SAM per unit length in the form

${\bar{S}}_{z}=\cos(\Theta_{P})\left(\frac{{\cal P}_{T}}{\omega
k_{z}}\right).$ (40)

The expression between the brackets has the dimensions of angular momentum per
unit length. Note that this is independent of $m$, which means that all higher
order modes have the invariant SAM which depends only on the Poincare angle
$\Theta_{P}$. The factor $\cos(\Theta_{P})$ takes the values $\pm 1$ at
$\Theta_{P}=0,\pi$ as for circularly-polarised modes and is zero for
$\Theta_{P}=\pi/2$, as for radially and azimuthally-polarised modes. Other
values of $\Theta_{P}$ concern elliptically-polarised modes.

###  III.6 AM desnity

The cycle-averaged angular momentum density is defined as follows:

${\mbox{\boldmath$\bar{j}$}}={\bf r}\times{\bar{\mbox{\boldmath$\pi$}}},$ (41)

where ${\bar{\mbox{\boldmath$\pi$}}}=\frac{1}{c^{2}}{\bar{\bf w}}$ is the
linear momentum density with ${\bar{\bf w}}=\frac{1}{2\mu_{0}}\Re[{\bf
E}^{*}\times{\bf B}]$ the energy density. Thus in order to proceed we have to
work out all the components of the Poynting vector. The vector product $[{\bf
E}^{*}\times{\bf B}]$ where the fields are ${\bf E}={\bf E}_{1}+{\bf E}_{2}$
and ${\bf B}={\bf B}_{1}+{\bf B}_{2}$ is

$[{\bf E}^{*}\times{\bf B}]=[{\bf E}_{1}^{*}\times{\bf B}_{1}]+[{\bf
E}_{2}^{*}\times{\bf B}_{2}]+[{\bf E}_{1}^{*}\times{\bf B}_{2}]+[{\bf
E}_{2}^{*}\times{\bf B}_{1}].$ (42)

We refer to the components of the different terms generically by ${\cal
P}_{\alpha\beta i}$ where $\alpha,\beta$ each take the values 1 and 2 and
$i=(x,y,z)$.

As an example to how evaluations proceed, we consider ${\cal P}_{11x}$. We
have

${\cal P}_{11x}=[{\bf E}_{1}^{*}\times{\bf
B}_{1}]_{x}=E_{1y}^{*}B_{1z}-E_{1z}^{*}B_{1y},$ (43)

which gives

$\displaystyle{\cal P}_{11x}$ $\displaystyle=$
$\displaystyle(-ck_{z})\left\\{{{\cal
F}^{(1)}}^{*}\left[i\left(\frac{\partial{\cal F}^{(1)}}{\partial
x}\right)+\left(\frac{\partial{\cal F}^{(1)}}{\partial y}\right)\right]+{\cal
F}^{(1)}\left[-i\left(\frac{\partial{\cal F}^{(1)}}{\partial
x}\right)^{*}+\left(\frac{\partial{\cal F}^{(1)}}{\partial
y}\right)^{*}\right]\right\\}.$ (44)

We obtain on substituting for the derivatives

${\cal P}_{11x}=-2ck_{z}|{\cal U}_{P}|^{2}\left({\tilde{\cal
F}}^{\prime}{\tilde{\cal F}}+\frac{m}{\rho}{\tilde{\cal
F}}^{2}\right)\sin\phi.$ (45)

Similar evaluations were carried out for all 16 terms in Eq.(42). It turns out
that all the 8 mixed term components arising from $[{\bf E}_{1}^{*}\times{\bf
B}_{2}]+[{\bf E}_{2}^{*}\times{\bf B}_{1}]$ are imaginary and so will not
contribute to the real part of of the overall Poynting vector. From these we
are able to evaluate the components of ${\bar{\bf j}}$. For example the
y-component ${\bar{j}}_{y}$ is given by

${\bar{j}}_{y}=\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\left\\{z({\cal
P}_{11x}+{\cal P}_{22x})-x({\cal P}_{11z}+{\cal P}_{22z})\right\\}.$ (46)

The x-component ${\bar{j}}_{x}$ together with the y-component ${\bar{j}}_{y}$,
constitute the transverse angular momentum density vector along $\hat{\phi}$

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mbox{\boldmath${\bar{j}}$}_{\bot}=\frac{\epsilon_{0}c^{2}k_{z}^{3}}{2\omega}\rho{\tilde{\cal
F}}^{2}}{\mbox{\boldmath$\hat{\phi}$}}$ (47)

Similar evaluations of the longitudinal (z-component) of the angular momentum
density ${\bar{j}}_{z}$ in the form

${\bar{j}}_{z}=\frac{\epsilon_{0}ck_{z}}{2\omega}\left\\{x({\cal
P}_{11y}+{\cal P}_{22y})-y({\cal P}_{11x}+{\cal P}_{22x})\right\\}.$ (48)

We obtain

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bar{j}}_{z}=\frac{\epsilon_{0}c^{2}k_{z}^{2}}{2\omega}\left\\{m{\tilde{\cal
F}}^{2}+\rho{\tilde{\cal F}}{\tilde{\cal
F}}^{\prime}\right\\}\cos\Theta_{P}}.$ (49)

The angular momentum density vector is therefore given by

${\mbox{\boldmath$\bar{j}$}}={{\bar{j}}}_{\bot}{\mbox{\boldmath$\hat{\phi}$}}+{\bar{j}}_{z}{\mbox{\boldmath$\hat{z}$}},$
(50)

where ${{\bar{j}}}_{\bot}$ is the magnitude of the transverse component, given
by Eq.(47), and ${\bar{j}}_{z}$ is the longitudinal component, given by
Eq.(49).

###  III.7 Variations of ${{\bar{j}}}_{\bot}$ and ${\bar{j}}_{z}$ with $\rho$

For illustration we now examine the distributions of the angular momentum
density components for a representative case, namely the order $m=3$ Poincaré
mode of the Laguerre-Gaussian type for which the mode function ${\tilde{\cal
F}}$ is given by Eq.(21). Figure (2) displays the variations of the SAM
density components with the radial coordinate while Fig.(3) displays the
corresponding angular momentum density variations with the radial coordinates.

Figure 3: Variations of the components of the angular momentum density
${\bar{\bf j}}$ (arbitrary units) with the radial coordinate $\rho$. The blue
curve represents the variations of the transverse component (Eq. 47) and the
yellow curve represents the variations of the longitudinal component (Eq. 49).
For illustration we have chosen to consider the order $m=2$ and the vector
mode for which $\Phi_{p}=0$ and $\cos\Theta_{P}=0.99$ of the order m=2
Poincaré sphere.

###  III.8 Integrated angular momentum density

The space integral of the angular momentum density over the x-y plane is
formally defined by

${\mbox{\boldmath$\bar{J}$}}=\int_{0}^{\infty}\rho\;d\rho\int_{0}^{2\pi}d\phi\left\\{{{\bar{j}}}_{\bot}{\mbox{\boldmath$\hat{\phi}$}}+{\bar{j}}_{z}{\mbox{\boldmath$\hat{z}$}}\right\\}={{\bar{J}}}_{\bot}{\mbox{\boldmath$\hat{\phi}$}}+{\bar{J}}_{z}{\mbox{\boldmath$\hat{z}$}}.$
(51)

The integral of the transverse density vanishes by virtue of the $\phi$
integral. We can then write

${{\bar{J}}}_{\bot}=0.$ (52)

The integral of the longitudinal term is

$\displaystyle{\bar{J}}_{z}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{0}c^{2}k_{z}^{2}}{2\omega}$ (53)
$\displaystyle\times$ $\displaystyle\int_{0}^{\infty}\rho
d\rho\int_{0}^{2\pi}d\phi\left\\{m{\tilde{\cal F}}^{2}+\rho{\tilde{\cal
F}}{\tilde{\cal F}}^{\prime}\right\\}\cos\Theta_{P}.$

The two different integrals involved in the evaluation of Eq.(53) are detailed
in Appendix B. The result is

${\bar{J}}_{z}=\frac{k_{z}w_{0}^{2}\pi{\cal
E}_{0}^{2}}{4\mu_{0}c}\left[\cos\Theta_{P}(m-1)\right],$ (54)

and on substituting for ${\cal E}_{0}$, we find

$\displaystyle{\bar{J}}_{z}$ $\displaystyle=$ $\displaystyle\left(\frac{{\cal
P}_{T}}{\omega c}\right)\left[\cos\Theta_{P}(m-1)\right].$ (55)

The factor $\left(\frac{{\cal P}_{T}}{\omega c}\right)$ has the dimensions of
angular momentum per unit length, as is the case with SAM. Note, however, that
this total angular momentum depends on the order $m$. The factor $(m-1)$
appearing in Eq.(55) is consistent with the observation that in cylindrical
coordinates the two terms in the polarisation vector in Eq.(3) have phase
functions $e^{i(m-1)\phi}$ and $e^{-i(m-1)\phi}$. The factor
$\cos(\Theta_{P})$ takes the values $\pm 1$ at $\Theta_{P}=0,\pi$ as for
circularly-polarised modes and is zero for $\Theta_{P}=\pi/2$, as for radially
and azimuthally-polarised modes. Other values of $\Theta_{P}$ concern
elliptically-polarised modes. For $m=0$ and $\cos\Theta_{P}=1$ we recover the
conventional result of ${\cal P}_{T}/(\omega c)$ appropriate for circularly
polarised Gaussian modes. For $m=1$, we recover the zero angular momentum
appropriate for pure radially and azimuthally-polarised vortex modes.

###  III.9 Orbital angular momentum OAM

Having derived expressions for the AM density $\bar{j}$ and the SAM density
${\bar{\bf s}}$, we consider the difference
$({\mbox{\boldmath$\bar{j}$}}-{\mbox{\boldmath$\bar{s}$}})$ as representing
OAM density ${\bar{\bf\ell}}$ Allen  _et al._ (1999); Crimin and Barnett
(2024). Thus in our case the OAM density components are obtainable by
subtracting the SAM density components, Eqs.(34) and (36), from the angular
momentum density components, Eqs.(47) and (49). We have

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mbox{\boldmath$\bar{\ell}$}}={\mbox{\boldmath$\bar{j}$}}-{\mbox{\boldmath$\bar{s}$}}}$
$\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(\frac{\epsilon_{0}c^{2}k_{z}}{2\omega}\right)\left\\{k_{z}^{2}\rho{\tilde{\cal
F}}^{2}-\frac{m{\tilde{\cal F}}^{2}}{\rho}-{\tilde{\cal
F}^{\prime}}{\tilde{\cal F}}\right\\}{\mbox{\boldmath$\hat{\phi}$}}}$ (56)
$\displaystyle+$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\cos\Theta_{P}\left(\frac{k_{z}\epsilon_{0}c^{2}}{2\omega}\right)\left\\{m{\tilde{\cal
F}}^{2}+\rho{\tilde{\cal F}}{\tilde{\cal F}}^{\prime}+{\tilde{\cal
F}}^{2}\right\\}{\mbox{\boldmath$\hat{z}$}}},$

and we note the two separate orbital angular momentum density components,
namely the azimuthal component and the longitudinal one, both of which are
seen to depend on the choice of $m$ as well as the Poincaré angle
$\Theta_{P}$. For illustration, we display in Fig.(4) the variations of the
OAM density components with the radial coordinate (in units of $w_{0}$).

Figure 4: Variations of the orbital angular momentum density (in arbitrary
units) with the radial coordinate $\rho$. The blue curve represents the
variations of the transverse component and the yellow curve the longitudinal
component (Eq. 56). For illustration we have chosen to consider the order
$m=2$ and the vector mode close to the north pole point for which
$\cos\Theta_{P}=0.99$.

###  III.10 OAM per unit length

The OAM per unit length is the space integral of the OAM density components
over a cross section of the mode. Clearly the azimuthal component of the OAM
component vanishes due to the angular integration. The longitudinal component
is then given by

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bar{L}}_{z}=\cos\Theta_{P}\left(\frac{\pi
k_{z}\epsilon_{0}c^{2}}{\omega}\right)\int_{0}^{\infty}\left\\{m{\tilde{\cal
F}}^{2}+\rho{\tilde{\cal F}}{\tilde{\cal F}}^{\prime}+{\tilde{\cal
F}}^{2}\right\\}\rho\;d\rho.}$ (57)

For a Laguerre-Gaussian mode for which the amplitude function is given by
Eq.(21) the integrals involved in the evaluation of Eq.(57) are detailed in
Appendix B. It is easy to see that the integrals of the last two terms in
Eq.(13) are ${\cal I}_{Q}$ and ${\cal I}_{P}$, which have the same magnitude,
but opposite signs. These terms now cancel identically and we obtain for the
OAM density per unit length

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\bar{L}}_{z}=m\cos\Theta_{P}\left(\frac{{\cal
P}_{T}}{\omega c}\right).}$ (58)

Thus for $m=0$ we have a Gaussian mode with zero OAM, so that the total
angular momentum is entirely due to optical spin. For $m\neq 0$ the OAM
increases with increasing $m$, with $\cos\Theta_{P}$ spanning the range
$(+1,-1)$ involving generally elliptical polarisations.

###  III.11 Optical helicity

A recently published brief account of the helicity of higher order Poincaré
modes appeared as a short communication by the current authors Babiker  _et
al._ (2024). That communication provided little details of the procedure
leading to the evaluations of the helicity density and its space integral as
the helicity per unit length. The formalism was also characterised by the use
of cylindrical polar coordinates to describe the fields. Only the results for
helicity density and helicity per unit length of the higher order modes for
which $p=0$ were presented in babiker2024. However, we feel that it is
necessary to discuss here the details of the evaluations specifically in
Cartesian coordinates for the general case of $m\geq 0$ and $p\neq 0$ as the
evaluations are both intricate and worthy of presentation using the Cartesian
coordinate system we have adopted throughout this paper. As we shall see the
results that emerge are more general and show that the inclusion of the radial
number $p>0$ enhances the super-chirality, but we shall also show that we
recover the special case $p=0$ discussed inBabiker  _et al._ (2024).

The cycle-averaged optical helicity density and chirality density are defined
generally in Eq.(14). There are thus contributions arising from the total
electric field ${\bf E}=({\bf E}_{1}+{\bf E}_{2})$ and magnetic field ${\bf
B}=({\bf B}_{1}+{\bf B}_{2})$, so that

$\displaystyle{\bar{\eta}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{c}{\omega^{2}}{\bar{\chi}}$ (59) $\displaystyle=$
$\displaystyle-\frac{\epsilon_{0}c}{2\omega}\Im{[{\bf(E_{1}+E_{2})}^{*}\cdot{\bf(B_{1}+B_{2})}]},$

where ${\bf E}_{i}$ and ${\bf B}_{i}$, with $i=1,2$ are as given in
Eqs.(LABEL:emfields). We focus on the helicity from which the chirality can be
determined using Eq.(59) and proceed to evaluate both the helicity density and
its space integral for a general higher order optical vortex mode. The four
terms arising from the expansion of Eq.(59) are evaluated separately and now
require as a first step expressions for the x and y-derivatives of ${\cal
F}^{(1)}$ and ${\cal F}^{(2)}$ in polar coordinates which are displayed in
Eqs.(15) to (18). It turns out that the sum ${\bf E}_{1}^{*}\cdot{\bf
B}_{2}+{\bf E}_{2}^{*}\cdot{\bf B}_{1}$ does not contribute an imaginary part
and only the two direct terms ${\bf E}_{1}^{*}\cdot{\bf B}_{1}+{\bf
E}_{2}^{*}\cdot{\bf B}_{2}$ contribute. As an example how the evaluations
proceed we consider the first dot product term. We have using the first and
second equations in Eq.(LABEL:emfields)

$\displaystyle[{\bf E_{1}}^{*}\cdot{\bf B_{1}}]$ $\displaystyle=$
$\displaystyle 2ick_{z}^{2}|{\cal
F}^{(1)}|^{2}+c\left\\{i\left|\frac{\partial{\cal F}^{(1)}}{\partial
x}\right|^{2}+i\left|\frac{\partial{\cal F}^{(1)}}{\partial
y}\right|^{2}-\left(\frac{\partial{\cal F}^{(1)}}{\partial
y}\right)^{*}\left(\frac{\partial{\cal F}^{(1)}}{\partial
x}\right)+\left(\frac{\partial{\cal F}^{(1)}}{\partial
x}\right)^{*}\left(\frac{\partial{\cal F}^{(1)}}{\partial y}\right)\right\\}.$
(60)

The next steps involve substituting for the x- and y-derivatives of ${\cal
F}^{(1)}$ using Eqs.(15) and (16). This leads from Eq.(60) to the result for
the first term in the helicity density

$\displaystyle\frac{1}{|{\cal U}_{P}|^{2}}\Im{[{\bf E_{1}}^{*}\cdot{\bf
B_{1}}]}$ $\displaystyle=$ $\displaystyle 2ck_{z}^{2}|{\tilde{\cal
F}}|^{2}+c\left\\{|{\tilde{\cal
F}}^{\prime}|^{2}+\frac{m^{2}}{\rho^{2}}|{\tilde{\cal F}}|^{2}\right\\}$ (61)
$\displaystyle+$ $\displaystyle c\frac{2m}{\rho}{\tilde{\cal
F}}^{\prime}{\tilde{\cal F}}.$

The second term $\Im{[{\bf E_{2}}^{*}\cdot{\bf B_{2}}]}$ follows the same
steps to obtain for the dot product

$\displaystyle\frac{1}{|{\cal V}_{P}|^{2}}\Im{[{\bf E_{2}}^{*}\cdot{\bf
B_{2}}]}$ $\displaystyle=$ $\displaystyle-\left(2ck_{z}^{2}|{\tilde{\cal
F}}|^{2}+c\left\\{|{\tilde{\cal
F}}^{\prime}|^{2}+\frac{m^{2}}{\rho^{2}}|{\tilde{\cal
F}}|^{2}\right\\}\right)$ (62) $\displaystyle-$ $\displaystyle
c\frac{2m}{\rho}{\tilde{\cal F}}^{\prime}{\tilde{\cal F}}.$

Thus we find

$\frac{1}{|{\cal U}_{P}|^{2}}\Im{[{\bf E_{1}}^{*}\cdot{\bf
B_{1}}]}=-\frac{1}{|{\cal V}_{P}|^{2}}\Im{[{\bf E_{2}}^{*}\cdot{\bf B_{2}}]}.$
(63)

After some algebra, the results (61) and (62), together with the use of
Eq.(5), lead to the final expression for the helicity density

${\bar{\eta}}({\bf
r})=\frac{\epsilon_{0}c^{2}}{4\omega}\cos{(\Theta_{P})}\left\\{2k_{z}^{2}|{\tilde{\cal
F}}_{m,p}|^{2}+|{\tilde{\cal F}}^{\prime}_{m,p}|^{2}+m^{2}\frac{|{\tilde{\cal
F}}_{m,p}|^{2}}{\rho^{2}}+2m\frac{{\tilde{\cal F}}_{m,p}^{\prime}{\tilde{\cal
F}}_{m,p}}{\rho}\right\\}.$ (64)

The first two terms of Eq.(64) are identifiable as the zero order ($m=0$)
helicity density for a general elliptical polarisation. The rest of terms are
the $m$-dependent higher-order terms.

Recall that the Poincaré angle $\Theta_{P}$ spans the range $\Theta_{P}=0$ to
$\Theta_{P}=\pi$ and within this range the Poincare function
$\cos(\Theta_{P})$ varies continuously from +1.0 ($\Theta_{P}=0$, which
corresponds to right-hand circular polarisation at the north pole), to -1.0;
$\Theta_{P}=\pi$ (left-hand circular polarisation at the south pole). Between
the two pole points where $0<\Theta_{P}<\pi$ we have elliptical polarisation.
However when $\Theta_{P}=\pi/2$ and $m=1$, we have radial and azimuthal
polarisation and we see that the helicity density vanishes at all points on
the equatorial circle in this case and for all higher order $m$.

The general polarisation state covers a continuous set of points on longitudes
on the surface of the order $m$ unit Poincar’e sphere. In order to specify the
helicity density we only need the order $m$, the angles
$(\Theta_{P},\Phi_{P})$ and the optical vortex amplitude function
${\tilde{\cal F}}$. Note that the general helicity density Eq.(64) shows no
dependence on $\Phi_{P}$. This means, for example, all points on a given
latitude intersecting a given longitude point have the same helicity density.

A special case is that for which $m=0$ and $\Theta_{P}=0$ or $\pi$, for which
all the $m$-dependent terms of Eq.(64) are zero. The remaining expression
corresponds to the helicity density of circularly-polarised general optical
vortex mode Babiker  _et al._ (2022). This is because in this case the overall
factor $\cos(\Theta_{P})=\pm 1$ can be identified as $\sigma=\pm 1$, as for
circular polarisation.

The helicity density at the general point ${\Theta_{P},\Phi_{P}}$ as given by
Eq.(64) and with $m\neq 0$ the $m$-dependent terms come into play for all
values of $\cos{(\Theta_{P})}$ in the range $(+1.0\;{\rm{to}}\;-1.0)$,
corresponding to elliptically polarised modes (including circular, linear, as
well as radial and azimuthal). In particular, for $m\geq 1$, as $\Theta_{P}$
increases from its value at the north pole, the function $\cos(\Theta_{P})$
vanishes at $\Theta_{P}=\pi/2$ for all points $\Phi_{P}$ on the equatorial
circle. It is only for the first order $m=1$ that this vanishing density
corresponds to pure radially-polarised optical vortex modes.

We may now evaluate the super-chirality properties of higher order for the
special case involving Laguerre-Gaussian modes. The amplitude function for
topological order $m$, radial number $p$ and waist $w_{0}$ is given by
Eq.(21). The results are shown in Fig. 5 in which the helicity density
variations are shown for different topological order $m$. Figure 5 displays
the helicity density in this special case with topological order values
$m=0,1,25,50,100$ and $150$ for $\Theta_{P}=0$. It is clear from Fig.5 that
for $m=0$ and $m=1$ the helicity density is maximum on axis $\rho=0$ and that
the case $m=1$ is more than double that for $m=0$, which makes the case $m=1$
super-chiral. The $m=1$ helicity density variation contrasts with the cases
$m\geq 2$ in which the helicity density vanishes at $\rho=0$ and the maxima
occur off-axis. For $m\geq 2$ the density is initially marginally larger than
the case $m=0$. It is concentrated off-axis $\rho>0$ and it initially
decreases with increasing $m$, but then increases with increasing $m$.

In Fig. 6 we focus on the helicity density variations of the lowest orders
$m=0$ and $m=1$ and the $p$ values $p=0$ and $p=1$ using the same parameters
as in Fig.5. We see that the maximum of the first order $m=1,p=0$, as in Fig.5
exceeds that of $m=0,p=0$, However, the case $m=1;p=1$ shows a much higher
value of the helicity density at $\rho=0$ as a main feature of this mode. This
observation confirms that this mode behaves like a Gaussian, as for $m=0$, and
is indicative of spin-to-orbit conversion such that $\sigma+m=0$. It follows
that since we have $m=1$, we must have $\sigma=-1$ corresponding to one of the
components forming the Poincaré polarisation
${\mbox{\boldmath$\epsilon$}}_{m=1,p}.$ This is consistent with our earlier
comment following Eq. (55) about the appearance of $(m-1)$ in the phase
functions and the form of axial angular momentum component ${\bar{J}}_{z}$.
When we consider a larger $w_{0}$, as in Fig.7 where $w_{0}=\lambda$, we find
that the on-axis helicity density for the case $p=0$; $m=1$ is lower than that
for the case $p=0$; $m=0$, in contrast with the corresponding case in Fig.6.
This is because the $p=0$; $m=1$ case is due to the longitudinal field
components becoming weaker when the beam width is larger. However, the
enhanced on-axis helicity densities in the cases $p=1;m=0$ and $p=1;m=1$ are
evidence of $p$-dependent and $m$-dependent contributions arising from the
longitudinal components as radial and azimuthal gradients of the transverse
fields, respectively.

###  III.12 Integrated helicity

We can evaluate the integral of the helicity density in Eq.(64)) over the
$x-y$ plane. First we note that the radial integral of all terms in the form
$\frac{{\tilde{\cal F}^{\prime}}{{\tilde{\cal F}}}}{\rho}$ are identically
zero for all mode functions which satisfy ${\tilde{\cal
F}}_{\\{m\\}}(0)=0={\tilde{\cal F}}_{\\{m\\}}(\infty)$. The integrals needed
to evaluate the helicity density per unit length are given in Appendix B for
the special case of Laguerre-Gaussian mode for which amplitude function is
given by Eq.(21).

Figure 5: Variations with the radial coordinate $\rho$ in units of $w_{0}$ of
the helicity density, Eq.(64), due to focused Poincare modes of orders
$m=0,1,25,50,100$ and $150$ with $p=0$ and $w_{0}=0.5\lambda$. Each of these
modes lies on the longitude $(\Theta_{P},\Phi_{P}=0)$, so that the values at a
given point on this longitude should be multiplied by $\cos{(\Theta_{P})}$.
Note that values on each curve are given relative to the maximum of the
Gaussian helicity density ($m=0,p=0$; blue solid curve) which is set to the
value of $1.0$. Superchirality for a given curve is said to occur when the
helicity density at a given radial position exceeds $1.0$. Figure 6:
Variations with the radial coordinate $\rho$ in units of $w_{0}$ of the
helicity density, Eq.(64), due to focused Poincaré modes for which $m=0,1$ and
$p=0,1$ with $w_{0}=0.5\lambda$. The modes lie on the longitude
$(\Theta_{P},\Phi_{P}=0)$, so that the values at any other given point on this
longitude should be as shown, but multiplied by $\cos{(\Theta_{P})}$.Note that
values on each curve are given relative to the maximum of the Gaussian
helicity density ($m=0,p=0$; blue solid curve) which is set to the value of
$1.0$. Superchirality for a given curve is said to occur when the helicity
density at a given radial position exceeds $1.0$. Figure 7: As in Fig.6, but
with a larger beam width at focus $w_{0}=\lambda$. Note the general reduction
in all cases of the helicity density due to this. But the peak helicity
density for the case of $m=1$; $p=1$ still exceeds that of our reference case
($m=0$; $p=0$), even for this larger beam width. See the main text for further
remarks on this figure.

We are finally led to the helicity per unit length at the focal plane  as

${\bar{\cal C}}_{m,p}={\cal
L}_{0}\cos{(\Theta_{P})}\left(1+\frac{2p+m+1}{k_{z}^{2}w_{0}^{2}}\right),$
(65)

where, as pointed out above, ${\cal L}_{0}={\cal P}_{T}/(\omega c)$ is a
constant for a fixed power ${\cal P}_{T}$ and we have substituted for ${\cal
A}_{0}$ using Eq.(75). The physical interpretation of the two terms within the
brackets is that the first term accounts for the helicity due to the
transverse fields which are formed as a superposition of two opposite
circularly polarised vortex modes. The second term accounts for an additional
helicity due to the longitudinal field components which arise from the non-
zero gradient of the transverse field components in the same manner in which
the Gouy phase arises (Feng and Winful (2001)) as appears in Eq. (20) where we
have the same $(2p+|m|+1)$ factor. Note that although the factor
$1/k_{z}^{2}w_{0}^{2}$ in Eq.(65) is typically small for $w_{0}^{2}\gg
1/k_{z}^{2}$, the higher order helicity for which $m\gg 1$ and/or $p\geq 1$
would ensure super-chirality for relatively large $w_{0}$.

##  IV 4\. Summary

The main mission of this paper has been to determine the intrinsic properties
of Poincaré higher order modes. We have pointed out the significance of these
modes for the processes of optical trapping of atoms and small particles and
also in the manipulation of chiral matter and in providing improved encoding
schemes for larger bandwidth optical quantum communications. Here their
intrinsic properties, as we emphasised, have previously neither been
considered, nor their values determined. We have argued that our approach
towards determining the intrinsic properties is most essential because it
takes into account the participation of the axial components of the twisted
light fields of the Poinaré modes. We have shown that proper determinations of
their angular momentum, both spin and orbital, their helicity and the
chirality require the inclusion of the longitudinal field components. We have
predicted significant enhancements of the values of the intrinsic properties
of these modes when compared with those due to the zero order optical modes.

Our first finding in this paper consists of determining the spatial
distributions of the spin angular momentum (SAM) density components, both
transverse and longitudinal of the higher order modes. However on integrating
the density components, we only find a non-zero longitudinal component which
is independent of $m$ . This means that all higher order modes have the same
SAM which depends only on the Poincare angle $\Theta_{P}$. The factor
$\cos(\Theta_{P})$ takes the values $\pm 1$ at $\Theta_{P}=0,\pi$ as for
circularly-polarised modes and is zero for $\Theta_{P}=\pi/2$, as for $m=1$
radially and azimuthally-polarised modes. Other values of $\Theta_{P}$ concern
elliptically-polarised modes.

Our second finding concerns the angular momentum (AM) density, which for
paraxial light must be the sum of the SAM density and the OAM density. We
obtained the OAM density by subtracting the SAM density from the AM density.
We then evaluated the space integrals of both the AM and the OAM and found
that they include new terms dependent on the order $m$ and on
$\cos(\Theta_{P})$. However, these expressions reduce to the known zero order
and first order results for any point on the surface of the respective order
Poincaré sphere.

Finally, we tackled the helicity and chirality of these higher order vortex
modes, obtaining a general expression for the helicity density and discussed
helicity for the relevant special lower order cases. In particular, for points
$(\pi/2,\Phi_{P})$ on the equatorial circle, the helicity (and so also
chirality) is always zero, as for pure radially and azimuthally-polarised
modes. The case $m=1,p=1$ displays the highest helicity density which is at
least double the $m=0$ helicity density. This means that this mode exhibits
super-chirality since it is enhanced relative to the helicity of an ordinary
(order $m=0$) circularly-polarised mode. For all other points on the surface
of the Poincaré sphere the variations of the helicity density display an
initial decrease with increasing $m$ followed by an increase. We have also
shown that a higher order $m$ Laguerre-Gaussian mode for which $m=1,p=1$ is a
strongly super-chiral vortex beam which is dominated by the vortex core at
$\rho=0$. We have found that other higher order Laguerre-Gaussian modes for
which $m>1,p>0$ have off-axis maximum helicity which is also super-chiral.
These results strongly indicate the existence of a highly desirable super-
chirality property of the higher order modes which, we suggest, is now ripe
for direct experimental investigation. There are diverse applications that can
be envisaged, including improved interactions with chiral matter and stronger
trapping and manipulation using optical spanners and tweezers, for example in
micro-fluidics and improved encoding schemes for higher bandwidth quantum
communications.

## Appendix A: Normalisation

The higher order vortex mode normalisation is related to the applied power
${\cal P}_{T}$ evaluated as the space integral over the beam cross-section of
the z-component of the Poynting vector. We have

${\cal
P}_{T}=\frac{1}{2\mu_{0}}\int_{0}^{2\pi}d\phi\int_{0}^{\infty}|\left\\{({\bf
E}_{1}+{\bf E}_{2})^{*}\times({\bf B}_{1}+{\bf B}_{2})\right\\}_{z}|\rho
d\rho.$ (66)

There are 4 terms, 2 direct and two mixed terms, but once again, we find that
the mixed terms cancel. Consider first the mixed terms. We have

$\displaystyle{\cal P}_{12z}$ $\displaystyle=$ $\displaystyle[{\bf
E}_{1}^{*}\times{\bf B}_{2}]_{z}$ (67) $\displaystyle=$ $\displaystyle
E_{1x}^{*}B_{2y}-E_{1y}^{*}B_{2x}$ $\displaystyle=$ $\displaystyle{\cal
U}_{P}^{*}{\cal
V}_{P}\left\\{(-ick_{z})(ik)-(ck_{z})(k_{z})\right\\}{\tilde{\cal F}}^{2}$
$\displaystyle=$ $\displaystyle 0.$

Similarly, we have

$\displaystyle{\cal P}_{21z}$ $\displaystyle=$ $\displaystyle[{\bf
E}_{2}^{*}\times{\bf B}_{1}]_{z}$ (68) $\displaystyle=$ $\displaystyle
E_{2x}^{*}B_{1y}-E_{2y}^{*}B_{1x}$ $\displaystyle=$ $\displaystyle{\cal
U}_{P}^{*}{\cal
V}_{P}\left\\{(-ick_{z})(ik_{z})-(-ck_{z})(-k_{z})\right\\}{\tilde{\cal
F}}^{2}$ $\displaystyle=$ $\displaystyle 0.$

Also the direct terms are as follows

$\displaystyle{\cal P}_{11z}$ $\displaystyle=$ $\displaystyle[{\bf
E}_{1}^{*}\times{\bf B}_{1}]_{z}$ (69) $\displaystyle=$ $\displaystyle
E_{1x}^{*}B_{1y}-E_{1y}^{*}B_{1x}$ $\displaystyle=$ $\displaystyle|{\cal
U}_{P}|^{2}\left\\{(-ick_{z})(ik_{z})+(ck_{z})(k_{z})\right\\}{\tilde{\cal
F}}^{2}$ $\displaystyle=$ $\displaystyle 2ck_{z}^{2}{\tilde{\cal F}}^{2}|{\cal
U}_{P}|^{2},$

and

$\displaystyle{\cal P}_{22z}$ $\displaystyle=$ $\displaystyle[{\bf
E}_{2}^{*}\times{\bf B}_{2}]_{z}$ (70) $\displaystyle=$ $\displaystyle
E_{1x}^{*}B_{1y}-E_{1y}^{*}B_{1x}$ $\displaystyle=$ $\displaystyle|{\cal
V}_{P}|^{2}\left\\{(-ick_{z})(ik_{z})+(ck_{z})(k_{z})\right\\}{\tilde{\cal
F}}^{2}$ $\displaystyle=$ $\displaystyle 2ck_{z}^{2}{\tilde{\cal F}}^{2}|{\cal
V}_{P}|^{2}.$

Collecting terms, we find, using the last identity in Eq.(5)

$\displaystyle\left\\{({\bf E}_{1}+{\bf E}_{2})^{*}\times({\bf B}_{1}+{\bf
B}_{2})\right\\}_{z}$ $\displaystyle=$ $\displaystyle 2ck_{z}^{2}{\tilde{\cal
F}}^{2}|(|{\cal U}_{P}|^{2}+|{\cal V}_{P}|^{2})$
$\displaystyle=ck_{z}^{2}{\tilde{\cal F}}^{2}.$ (71)

Then Eq.(66) yields

${\cal P}_{T}=\frac{\pi ck_{z}^{2}}{\mu_{0}}{\cal I}_{P},$ (72)

where

${\cal I}_{P}=\int_{0}^{\infty}{\tilde{\cal F}}^{2}\rho\;d\rho,$ (73)

so we can write

${\cal I}_{P}=\frac{\mu_{0}{\cal P}_{T}}{\pi ck_{z}^{2}}.$ (74)

This result applies to any optical vortex mode characterised by an amplitude
function ${\tilde{\cal F}}$. The integral ${\cal I}_{P}$ is evaluated in
Appendix B for the special case of a vector mode involving Laguerre-Gaussians
and so we can now evaluate the normalisation constant ${\cal A}_{0}$ using
Eqs.(74) and (77). We then have

${\cal A}_{0}^{2}=\frac{4\mu_{0}{\cal P}_{T}}{\pi ck_{z}^{2}w_{0}^{2}}.$ (75)

## Appendix B: Four Integrals

There are four integrals that require evaluation for the special case of a
Laguerre-Gaussian mode for which ${\tilde{\cal F}}$ is given by Eq.(21). These
are

$\displaystyle{\cal I}_{P}$ $\displaystyle=\int_{0}^{\infty}{\tilde{\cal
F}}^{2}\rho\;d\rho,$ (76) $\displaystyle{\cal I}_{Q}$
$\displaystyle=\int_{0}^{\infty}\rho^{2}{\tilde{\cal F}^{\prime}}{\tilde{\cal
F}}\;d\rho,$ $\displaystyle{\cal I}_{R}$
$\displaystyle=\int_{0}^{\infty}\frac{1}{\rho}{\tilde{\cal F}}^{2}d\rho,$
$\displaystyle{\cal I}_{S}$ $\displaystyle=\int_{0}^{\infty}|{\tilde{\cal
F}^{\prime}}|^{2}\rho\;d\rho.$

Substituting for ${\tilde{\cal F}}$ from Eq.(21) and using the variable
$x=2\rho^{2}/w_{0}^{2}$ we have for ${\cal I}_{P}$

${\cal I}_{P}={\cal
A}_{0}^{2}\frac{w_{0}^{2}}{4}\frac{(p+|m|)!}{p!}\int_{0}^{\infty}e^{-x}x^{|m|}[L^{|m|}_{p}(x)]^{2}dx=\frac{{\cal
A}_{0}^{2}w_{0}^{2}}{4},$ (77)

where the x-integral is a standard integral of the associated Laguerre-
functions.

Consider next the evaluation of ${\cal I}_{Q}$. We have on substituting for
${\tilde{\cal F}}$ from Eq.(21) and using the variable $x=2\rho^{2}/w_{0}^{2}$

${\cal I}_{Q}=\frac{w_{0}^{2}}{2}\int_{0}^{\infty}x{\tilde{\cal
F}}(x){\tilde{\cal F}^{\prime}}(x)dx,$ (78)

where

${\tilde{\cal F}}(x)={\cal
A}_{0}^{2}\sqrt{\frac{p!}{(p+|m|)!}}x^{|m/2|}e^{-x/2}L^{|m|}_{p}(x).$ (79)

Integrating by parts in Eq.(78) we have

$\displaystyle{\cal I}_{Q}$ $\displaystyle=$
$\displaystyle\frac{w_{0}^{2}}{2}\int_{0}^{\infty}\frac{x}{2}\left(\frac{d{\tilde{\cal
F}}^{2}(x)}{dx}\right)dx$ (80) $\displaystyle=$
$\displaystyle\frac{w_{0}^{2}}{4}\left\\{x{\tilde{\cal
F}}(x)^{2}\right\\}_{0}^{\infty}-\frac{w_{0}^{2}}{4}\int_{0}^{\infty}{\tilde{\cal
F}}^{2}(x)dx.$

The first term yields zero at both limits and ${\cal I}_{Q}$ is given by the
second term. On substituting for ${\tilde{\cal F}}(x)$ using Eq.(79), we have

$\displaystyle{\cal I}_{Q}$ $\displaystyle=$
$\displaystyle-\frac{w_{0}^{2}}{4}{\cal
A}_{0}^{2}\frac{p!}{(p+|m|)!}\int_{0}^{\infty}x^{|m|}e^{-x}[L^{|m|}_{p}(x)]^{2}dx$
(81) $\displaystyle=$ $\displaystyle-\frac{w_{0}^{2}}{4}{\cal
A}_{0}^{2}=-{\cal I}_{P}.$

Next we consider ${\cal I}_{R}$. Substituting for ${\tilde{\cal F}}$ from
Eq.(21) and using the variable $x=2\rho^{2}/w_{0}^{2}$ we have for ${\cal
I}_{R}$

$\displaystyle{\cal I}_{R}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{1}{\rho}{\tilde{\cal F}}^{2}d\rho$ (82)
$\displaystyle=$
$\displaystyle\frac{p!}{2(p+|m|)!}\int_{0}^{\infty}x^{|m|-1}e^{-x}[L^{|m|}_{p}(x)]^{2}dx$
$\displaystyle=$ $\displaystyle{\cal A}_{0}^{2}\frac{1}{2|m|}.$

Finally we consider ${\cal I}_{S}$. On substituting for the first derivative
${\tilde{\cal F}}^{\prime}$, this integral splits into a number of terms which
are then evaluated separately. There are cancellations between the integrals
of those terms and the result turns out to be

$\displaystyle{\cal I}_{S}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\tilde{\cal F}}^{\prime 2}\rho
d\rho=\frac{2p+1}{2}{\cal A}_{0}^{2}.$ (83)

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