# Angular momentum transfered by the field of a moving point charge

Vladimir Epp  Department of Theoretical Physics, Tomsk State Pedagogical
University, 634061 Tomsk, Kievskaya 60, Russia<EMAIL_ADDRESS>Ulyana
Guselnikova  Department of Theoretical Physics, Tomsk State Pedagogical
University, Russia, 634061 Tomsk, Kievskaya 60, Russia.  Irina Kamenskaya
General Physics Department, Tomsk State Pedagogical University, Russia, 634061
Tomsk, Kievskaya 60, Russia.

###### Abstract

The flux of angular momentum of electromagnetic field of an arbitrarily moving
point charge is investigated. General equations are obtained for the transfer
of angular momentum at arbitrary distance from the charge, and corresponding
equations in the far-field approximation. An explicit expression is obtained
for the flux of angular momentum in the wave zone in terms of coordinates,
velocity, and acceleration of the charge. The torque is calculated, that would
act on an object if it absorbed all the radiation incident on it. It is shown
that this torque is proportional to the curl of the stress tensor of the
electromagnetic field; in the far field approximation the torque is
proportional to the curl of the Poynting vector.

orbital angular momentum; torque; electromagnetic field; charged particle

##  I Introduction

It is well known that electromagnetic field of a point charge carries angular
momentum [1]. The angular momentum is usually devided into a spin part
associated with polarization and an orbital part associated with helical wave
front. A light beam with a helical wave front is usually referred to as a
twisted light. The first theoretical and experimental research of twisted
light or vortex radiation, were devoted to laser radiation modified by
astigmatic optical system, numerically computed holograms [2, 3], or
microscopic spiral phase plates. These publications have stimulated extensive
studies of vortex optical beam. Further extensive bibliography on the history
in this field can be found in the recent review [4]. Interest in vortex
radiation quickly spread over different areas of physics: transfer of
information, interaction with atoms, high-energy particle collision and
radiation processes. The vortex light beams have opened a wide range of
applications, such as spatial optical trapping of atoms or microscopic
objects, phase-contrast microscopy, and nano- or micro-scale physics [4, 5].
Various methods are being developed to register the orbital angular momentum
of radiation [6, 7].

High energy photons carrying angular momentum can be emitted by the vortex
beams of charged particles. In recent years attention to the X-ray vortices
produced by high energy particles has been boosted by the interest to
microscopy and spectroscopy in the atomic and nanometric scale. Radiation in
the X-ray range carrying the angular momentum was obtained by converting an
x-ray beam [8] and by use of a helical undulator [9]. Various schemes of
twisted photon beam production in undulators [10, 11, 12] and free electron
lasers [13] have been proposed. Cherenkov radiation and transition radiation
emitted by vortex electrons was studied theoretically [14, 15, 16]. Radiation
of high energy charged particles channeled in the solid and liquid crystals
has been studied theoretically in recent papers [17, 18, 19, 20]. The X-ray
vortex radiation has found numerous applications both in classical and quantum
optics condensed matter, high energy physics, optics, etc. (see the review
[21] and references therein).

Despite active experimental research, there is a lack of theoretical studies
on transfer of angular momentum by the field of an arbitrarily moving charge.
The available theoretical papers in this area cover the angular momentum of
radiation only in some special cases, such as synchrotron radiation, radiation
in crystals, or polarization radiation.

This paper aims to fill this gap. In section II we obtain a general expression
for the flux of angular momentum through a unit area in terms of the stress
tensor of the field of a point charge. In section III, we study the flux of
angular momentum in the far-field approximation. Since the angular momentum
essentially depends on the specific coordinate system, we consider two cases –
when the coordinate origin is relative far from the charge, and when the
distance between the charge and the coordinate origin is much less than the
distance to the observation point. Approximate expressions for the flux of
angular momentum of a nonrelativistic charge are obtained in section IV.
Section V is devoted to study of the torque exerted on an area element due to
the electromagnetic field. The obtained expressions were applied to calculate
the angular momentum flux and the torque in the field of a rotating dipole in
section VI. Finally, section VII is devoted to discussion of the results
obtained.

##  II The flux of angular momentum in the field of a point charge

The angular momentum tensor $L^{\mu\nu}$ of the electromagnetic field is
defined by the energy-momentum tensor $T^{\mu\sigma}$ as [22]

$\displaystyle
L^{\mu\nu}=\int(x^{\mu}{T}^{\nu\sigma}-x^{\nu}T^{\mu\sigma}){\rm
d}S_{\sigma},$ (1)

where ${\rm d}S_{\sigma}$ is the vector equal in magnitude to the area of a
hypersurface element and normal to this element. The spatial components of the
energy-momentum tensor form a three-dimensional stress tensor

$\sigma_{ik}=\frac{1}{4\pi}\left[-E_{i}E_{k}-H_{i}H_{k}+\frac{1}{2}\delta_{ik}(E^{2}+H^{2})\right],$

with $E_{i}$ and $H_{i}$ being the components of electric and magnetic fields,
respectively, $\delta_{ik}$ is the Kronecker symbol. The time components
determine the energy and momentum density of the electromagnetic field

$T^{00}=\frac{E^{2}+H^{2}}{8\pi},\quad
T^{0i}=\frac{1}{c}P_{i},\quad\bm{P}=\frac{c}{4\pi}(\bm{E}\times\bm{H}),$

$\bm{P}$ is the Poynting vector.

Passing to three-dimensional notation, we introduce a three-dimensional
angular momentum vector with components $L_{i}=\frac{1}{2}e_{ijk}L^{jk}$,
$e_{ijk}$ is the unit antisymmetric symbol. The flux of the $i$-th component
of the vector $\bm{L}$ through the unit area orthogonal to the $k$-th axis is
determined by the three-dimensional tensor [22, 23]

$\displaystyle
g_{ik}=e_{ijm}x_{j}\sigma_{mk}=\frac{1}{4\pi}\left[-(\bm{r}\times\bm{E})_{i}E_{k}-(\bm{r}\times\bm{H})_{i}H_{k}+\frac{1}{2}(E^{2}+H^{2})e_{ijk}r_{j}\right].$
(2)

The flux of the angular momentum of the field through an arbitrarily oriented
area $\bm{{\rm d}s}$ is

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm
d}t}=\frac{1}{4\pi}\left[-(\bm{r}\times\bm{E})(\bm{E}\,\bm{{\rm
d}s})-(\bm{r}\times\bm{H})(\bm{H}\,\bm{{\rm
d}s})+\frac{1}{2}(E^{2}+H^{2})(\bm{r}\times\bm{{\rm d}s})\right].$ (3)

If the area $\bm{{\rm d}s}$ is orthogonal to the vector $\bm{r}$, then the
flux is

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm
d}t}=\frac{r}{4\pi}\left[(\bm{E}\times\bm{n})(\bm{E}\,\bm{n})+(\bm{H}\times\bm{n})(\bm{H}\,\bm{n})\right]{\rm
d}s.$ (4)

Here $\bm{n}=\bm{r}/r$. In the spherical coordinate system
$r,\,\theta,\,\varphi$ with unit vectors
$\bm{e}_{r},\bm{e}_{\theta},\bm{e}_{\varphi}$ the angular momentum flux in
radial direction is given by

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}t{\rm
d}s}=r\left(\sigma_{12}\bm{e}_{\varphi}-\sigma_{13}\bm{e}_{\theta}\right).$
(5)

The electric and magnetic fields of an arbitrary moving charge at a point
$\bm{r}$ and at a time moment $t$ are [22]

$\displaystyle\bm{E}$
$\displaystyle=\bm{E}_{1}+\bm{E}_{2},\quad\bm{H}=(\bm{R}\times\bm{E})/R,$ (6)
$\displaystyle\bm{E}_{1}$
$\displaystyle=\frac{eR^{2}\bm{\kappa}}{c(R-\bm{\beta}\bm{R})^{3}},\quad\bm{E}_{2}=\frac{e(1-\beta^{2})(\bm{R}-R\bm{\beta})}{(R-\bm{\beta}\bm{R})^{3}},$
(7) $\displaystyle\bm{\kappa}$
$\displaystyle=[\bm{R}\times[(\bm{R}-R\bm{\beta})\times\bm{\dot{\beta}}]]/R^{2},$

where $\bm{R}=\bm{r}-\bm{r}^{\prime}$ is the vector from the charge position
to the point $\bm{r}$, $\bm{r}^{\prime}=\bm{r}^{\prime}(t^{\prime})$ is the
charge position at the retarded time moment $t^{\prime}=t-R/c$, $c$ is the
speed of light, $e$ is the charge,
$\bm{\beta}=\bm{v}/c,\,\bm{v}=\bm{v}(t^{\prime})$ is the particle velocity,
and the dot denotes the time derivative.

##  III Flux of angular momentum in the far field approximation

Eqs. (2) – (7) determine the flux of angular momentum of the electromagnetic
field at any distance from the charge. To find the angular momentum carried by
the radiation of the charge, we obtain approximate equations for the angular
momentum at large distance. The field $\bm{E}_{1}$ decreases with distance as
$1/R$, and the field $\bm{E}_{2}$ as $1/R^{2}$. At large distances from the
charge, where $R\gg c\dot{\beta}$, the electric field $\bm{E}_{1}$ and the
corresponding magnetic field prevail. Therefore, when calculating the
intensity of radiation we can neglect the field $\bm{E}_{2}$.

Calculation of the angular momentum of radiation has one important difference
from the calculation of the intensity of radiation – the angular momentum
depends significantly on the choice of the coordinate system. When calculating
the intensity of radiation, two conditions are usually assumed to be
satisfied: i) $R\gg c\dot{\beta}$, then the field $\bm{E}_{2}$ can be
neglected, and ii) $R,r\gg r^{\prime}$ (see Fig. 1) and then you can put
$\bm{r}\approx\bm{R}$.

Figure 1: Notations used.

However, in practice, there are cases when the second condition is not met.
For example, when calculating the intensity of synchrotron radiation, it is
convenient to choose the origin at the center of a circular orbit, but
measurements are carried out at a distance comparable to or even less than the
orbital radius, as, for example, in Fig. 2. Nevertheless, the approximation i)
$R\gg c\dot{\beta}$ (far field or wave zone approximation) is sufficient to
calculate the radiation intensity.

Figure 2: Synchrotron radiation. The main part of the angular momentum
transferred to the target is due to the radiation pressure $\bm{g}$ on the
target which causes the torque relative to the $z$ axis.

A completely different situation arises when calculating the angular momentum
of radiation. Two substantially different situations should be distinguished
here: i) the target is in the wave zone ($R\gg c\dot{\beta}$), but the
distances $R$ and $r^{\prime}$ are comparable and ii) the target is in the
wave zone and $R,r\gg r^{\prime}$. Let’s consider these cases separately.

###  III.1 Angular momentum in the wave zone when $r^{\prime}\sim r,R$

In this case, we can neglect the field $\bm{E}_{2}$ in Eq. (4), and put
$\bm{E}\approx\bm{E}_{1}$. Then the vectors of the electric and magnetic
fields are mutually orthogonal and orthogonal to the direction of radiation
$\bm{N}=\bm{R}/R$. If we put in Eq. (4)
$\bm{H}=\bm{N}\times\bm{E},\,(\bm{E}\bm{N})=0$, then it can be transformed to
the form

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}t{\rm
d}s}=\frac{r}{4\pi}E^{2}(\bm{n}\times\bm{N})(\bm{n}\bm{N}){\rm
d}s=\frac{1}{c}(\bm{r}\times\bm{P})(\bm{N}\bm{n}),$ (8)

Thus, if the distances $r^{\prime},\,r,\,R$ are comparable in magnitude, then
the flux of angular momentum in the radiation field can be considered as a
mechanical torque due to the radiation pressure. The multiplier
$(\bm{n}\bm{N})$ in Eq. (8) takes into account that the area $\bm{\
ds}=\bm{n}{\rm d}s$ is oriented at an angle to the direction of radiation
$\bm{N}$. In this case, the angular momentum of radiation can be definitely
considered as the orbital angular momentum.

###  III.2 Angular momentum in the wave zone when $R,r\gg r^{\prime}$

Let us consider the case when the charge moves in a region whose dimensions
are much less than the distance $R$ and we are interested in the angular
momentum of the field relative to some point lying in the region of motion of
the charge ($r^{\prime}\ll R$). Then the scalar product $(\bm{E}\,\bm{n})$ in
the Eq. (4) becomes small and decreases with distance as $1/R^{2}$. In this
case, the contribution of the $\bm{E}_{2}$ field should also be taken into
account. In the limit $r\to\infty$, Eq. (4) takes the form

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm
d}t}=\frac{r^{3}}{4\pi}\left[(\bm{E}_{1}\times\bm{n})(\bm{E}_{2}\,\bm{n})+(\bm{H}_{1}\times\bm{n})(\bm{H}_{2}\bm{n})\right],$
(9)

where $\bm{H}_{i}=(\bm{R}\times\bm{E}_{i})/R$. Here each term decreases as
$1/r^{3}$ as $r\to\infty$.

This expression can also be expressed in terms of the Poynting vector. If we
make the substitution $\bm{H}_{1}=\bm{n}\times\bm{E}_{1}$ in Eq. (9), then we
get

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}s\,{\rm
d}t}=\frac{r}{4\pi}\left[\bm{E}_{1}(\bm{H}_{2}\bm{n})-\bm{H}_{1}(\bm{E}_{2}\bm{n})-\bm{n}(\bm{H}_{2}\bm{n})(\bm{E}_{1}\bm{n})\right].$
(10)

The first two terms decrease with distance as $1/r^{3}$, and the last term as
$1/r^{4}$. Therefore, it can be neglected. On the other hand, in the
considered approximation

$\displaystyle\bm{n}\times\bm{P}=\frac{c}{4\pi}[\bm{E}_{1}(\bm{n}\bm{H}_{2})-\bm{H}_{1}(\bm{n}\bm{E}_{2})].$
(11)

Hence, the flux of angular momentum in the wave zone in approximation $R,r\gg
r^{\prime}$ can be calculated through the Poynting vector

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}s\,{\rm
d}t}=\frac{1}{c}(\bm{r}\times\bm{P}).$ (12)

In the approximation $r^{\prime}\ll r$, this equation. in essence. coincides
with Eq. (8), since in the expansion of the scalar product $(\bm{N}\bm{{\rm
d}s})$ in powers of $r^{\prime}/r$ the first term is unit. As expected, the
flux of angular momentum in the wave zone of a point charge is described by
the well-known equation for the flux of angular momentum of electromagnetic
radiation [1, 24].

The flux of the modulus of the angular momentum of radiation can be related to
the radiation intensity. Let us denote by $\alpha$ the angle between the
vectors $\bm{n}$ and $\bm{P}$, and by $I$ – the radiation intensity (energy
per unit time). Then

$\displaystyle\frac{{\rm d}|\bm{L}|}{{\rm d}s\,{\rm
d}t}=\frac{1}{c}r\sin\alpha\frac{{\rm d}I}{{\rm d}s}.$ (13)

Let us find an explicit expression for the angular momentum flux in terms of
the charge position, its velocity and acceleration. Substituting
$\bm{R}=\bm{r}-\bm{r}^{\prime}$ into Eqs. (6), (7) and (9), and keeping the
largest term in the expansion in powers of $1/r$, we obtain

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm d}t}=\frac{e^{2}}{4\pi
c(1-\bm{\beta}\bm{n})^{5}}\left[(1-\beta^{2})(\bm{\kappa}\times\bm{n})+\frac{\kappa^{2}(\bm{r}^{\prime}\times\bm{n})}{c(1-\bm{\beta}\bm{n})}\right].$
(14)

Now, the vector $\bm{\kappa}$ contains only the main term of the expansion in
powers of $1/r$:

$\displaystyle\bm{\kappa}=[\bm{n}\times[(\bm{n}-\bm{\beta})\times\bm{\dot{\beta}}]].$
(15)

This equation determines the angular distribution of the angular momentum
carried by radiation of an arbitrarily moving charge. The first term in Eq.
(14) does not depend on the choice of the origin of coordinate system, while
the second one linearly depends on the position vector of the charge. In this
sense, the first term can be associated with the spin of radiation, and the
second one with the orbital angular momentum. The classical theory, in
principle, does not distinguish the spin and orbital parts of the total
angular momentum. Nevertheless, there is an active discussion on the issue of
dividing the angular momentum of electromagnetic field into orbital and spin
parts. See, for example, [25, 10, 26] and references therein. We would not
plunge into this discussion here. Therefore, we denote the studied total
angular momentum just as the angular momentum of radiation.

##  IV Non-relativistic approximation

In the non-relativistic approximation $\beta\ll 1$. Assuming in Eq. (14)
$\beta=0$ and taking into account that for $\beta=0$ the vector product
$(\bm{\kappa}\times\bm{n})=(\bm{n}\times\bm{\dot{\beta}})$, in the first
approximation in $\beta$ we have

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm d}t}=\frac{e^{2}}{4\pi
c}(\bm{n}\times\bm{\dot{\beta}}).$ (16)

Hence, the angular momentum of the radiation is orthogonal to the vectors
$\bm{n}$ and $\bm{\dot{\beta}}$. In the spherical coordinate system
$(r,\theta,\varphi)$ with the $z$ axis directed along $\bm{\dot{\beta}}$ and
the polar angle $\theta$, there remains only one component of the vector
$\bm{L}$ corresponding to the azimuth angle $\varphi$

$\displaystyle\frac{{\rm d}L_{\varphi}}{{\rm d}\Omega{\rm
d}t}=\frac{e^{2}\dot{\beta}}{4\pi c}\sin\theta.$ (17)

Obviously, if a particle moves within a limited region of space, then the
angular momentum transferred by the particle’s field is time dependent. The
linear dependence on $\dot{\beta}(t)$ vanishes the average angular momentum to
zero. Actually, Eq. (17) describes the exchange of angular momentum between
the near and far zones and, therefore, does not represent emission of angular
momentum. Moreover, the angular momentum of the radiation integrated over the
solid angle is in this approximation zero.

In order to calculate the nonzero mean value of the angular momentum flux, it
is necessary to take into account the next term in the expansion in $\beta$.
As a result of the expansion up to $\beta^{2}$, we obtain

$\displaystyle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm d}t}=\frac{e^{2}}{4\pi
c}\left[(\bm{n}\times\bm{\dot{\beta}})(1+4\bm{\beta}\bm{n})+(\bm{n}\times\bm{\beta})(\bm{n}\bm{\dot{\beta}})\right].$
(18)

Let us find the time-average angular momentum flux. When averaging over time
$t$, one should keep in mind that the right-hand side of the last expression
depends on $t^{\prime}$, and ${\rm d}t=(1-\bm{\beta}\bm{n}){\rm d}t^{\prime}$.
So

$\displaystyle\left\langle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm
d}t}\right\rangle=\frac{e^{2}}{4\pi
c}\frac{1}{T}\int\limits_{0}^{T}\left[(\bm{n}\times\bm{\dot{\beta}})(1+4\bm{\beta}\bm{n})+(\bm{n}\times\bm{\beta})(\bm{n}\bm{\dot{\beta}})\right](1-\bm{\beta}\bm{n}){\rm
d}t^{\prime}.$ (19)

The average of the term linear in $\bm{\dot{\beta}}$ is zero, and averaging
the remaining term gives

$\displaystyle\left\langle\frac{{\rm d}\bm{L}}{{\rm d}\Omega{\rm
d}t}\right\rangle=\frac{e^{2}}{4\pi c}\left\langle
3(\bm{n}\times\bm{\dot{\beta}})(\bm{\beta}\bm{n})+(\bm{n}\times\bm{\beta})(\bm{n}\bm{\dot{\beta}})\right\rangle.$
(20)

Integrating this expression over the solid angle, we obtain the total angular
momentum emitted per unit time

$\displaystyle\left\langle\frac{{\rm d}\bm{L}}{{\rm
d}t}\right\rangle=\frac{2e^{2}}{3c}\left\langle\bm{\beta}\times\bm{\dot{\beta}}\right\rangle.$
(21)

The vector $(\bm{\beta}\times\bm{\dot{\beta}})$ indicates the direction of the
instantaneous mechanical angular momentum of the particle. The last expression
up to sign coincides with the well-known equation for the loss of angular
momentum of a charged particle due to radiation friction [22].

##  V Torque exerted by the field

By definition, the angular momentum of electromagnetic field depends on the
choice of the coordinate system. Under certain conditions, as discussed in
section III.1, radiation pressure can make a significant contribution to
angular momentum. For example, the orbital angular momentum of a photon of
synchrotron radiation in an accelerator (Fig. 2) is equal to the product of
the photon momentum by the orbital radius $a$

$L=\frac{\hbar\omega}{c}a\sim\gamma^{3}\hbar,$

where $\omega\sim\gamma^{3}c/a$ is the photon frequency,
$\gamma=(1-\beta^{2})^{-1/2}$ is the relativistic factor. In modern
accelerators, this value can reach gigantic values. In this case, as in many
others, the angular momentum of the radiation relative to the geometric center
of the trajectory is of interest. But if we investigate the radiation of an
unknown source, then the position of this geometric center is not determined.
Or the source of radiation is so far away that the entire area of radiation
shrinks to a point, as is usually the case in astronomy. In these cases, Eq.
(14) has little sense. Only the integral over the solid angle is important,
because it determines the rate of loss of angular momentum by the charge due
to radiation friction.

However, the ‘vortex radiation’ has a property that can theoretically be
measured without being bound to any particular coordinate system. To date, a
large number of experiments have been carried out in which a vortex laser beam
drives the microparticles into rotation [27, 28, 29, 30]. The first
experiments on the capture of particles in traps and their rotation were
carried out at the end of the last century [31, 32] and are still being
carried out [28, 29, 30]. A detailed description of these works is given in
recent review [33].

Obviously, the component of angular momentum parallel to the normal to an area
is not transferred through this area, since the diagonal elements of the
tensor $g_{ij}$ are zero. However, if the field of the stress tensor is not
uniform and the area is of finite diminsion, this is quite possible.

Consider a small area $S$ oriented orthogonel to the radius vector passing
through the center of the area. The smallness of the area means that its
dimensions are much less than the distance to the origin of coordinate system
relative to which the angular momentum is calculated. Let’s denote the
coordinate of the center of the area through $\bm{r}_{0}$ as shown in Fig. 3.

Figure 3: The flux of the normal component of the angular momentum of the
field through the area $S$.

The unit normal to the area is $\bm{n}_{0}=-\bm{r}_{0}/r_{0}$. Let’s find the
torque acting on the area due to the total flux of the angular momentum
through the area. The physical meaning of the components of the tensor
$\sigma_{ij}$ is that they represent forces acting per unit area. The diagonal
elements represent pressure, and the off-diagonal element $\sigma_{ij}$ is
shear acting in the $i$-th direction on a unit area with the $j$-th normal.

Fig. 4 schematically shows the lines of the shear force vector $\sigma_{i1}$
on the area orthogonal to the radius vector $\bm{r}_{0}$. If the field of
shears is not uniform, they produce a torque acting on the area.

Figure 4: Lines of force of the vector $g_{i1}$ on the area $S$.

The flux of the normal component of the angular momentum through the area is
given by integral

$\displaystyle\frac{{\rm d}L_{n}}{{\rm d}t}=\int n_{0i}g_{i1}(\bm{r}){\rm
d}s=\int n_{0i}e_{ijm}x_{j}\sigma_{m1}(\bm{r}){\rm d}s.$ (22)

Here $g_{ij}(\bm{r})$ is the tensor (2), index 1 corresponds to the radial
coordinate. Obviously, the flux of the normal component of the angular
momentum through the center of the area is zero, since
$n_{0i}e_{ijm}x_{0j}\sim\bm{n}_{0}\times\bm{r}_{0}=0$. However, as can be seen
from Fig. 3, at a distance from the center, the cross product
$\bm{n}_{0}\times\bm{r}$ is no longer zero.

Let us expand the components of the tensor $\sigma_{m1}(\bm{r})$ in a Taylor
series in the vicinity of the center. Further on, we use the spherical
coordinate system $(x_{1},x_{2},x_{3})=(r,\theta,\varphi)$. Let us introduce
on the area mutually orthogonal coordinates $u,\,v$ such that the coordinate
lines of $u$ and $v$ coincide with the coordinate lines of $\theta$ and
$\varphi$, respectively. Then

$\sigma_{m1}(\bm{r})=\sigma_{m1}(\bm{r}_{0})+\left(u\frac{\partial\sigma_{m1}(\bm{r})}{\partial
u}\bigg{\rvert}_{0}+v\frac{\partial\sigma_{m1}(\bm{r})}{\partial
v}\bigg{\rvert}_{0}\right)$

The vertical bar means that the value of derivatives is taken at the center.
Next we substitute this in Eq. (22). The vector $\bm{n}_{0}$ has coordinates
$n_{0i}=(n_{01},-u/r,-v/r).$ For the sake of simplicity, assume that the area
is a disk. Using polar coordinates on the surface of the disk

$u=\rho\cos\psi,\quad v=\rho\sin\psi$

we transform the integral (22) to the form

$\displaystyle\frac{{\rm d}L_{r}}{{\rm
d}t}=\pi\left(\frac{\partial\sigma_{31}(\bm{r})}{\partial
u}\bigg{\rvert}_{0}-\frac{\partial\sigma_{21}(\bm{r})}{\partial
v}\bigg{\rvert}_{0}\right)\int\rho^{3}{\rm
d}\rho=\frac{1}{4\pi}S^{2}\left(\frac{\partial\sigma_{31}(\bm{r})}{\partial
x_{2}}\bigg{\rvert}_{0}-\frac{\partial\sigma_{21}(\bm{r})}{\partial
x_{3}}\bigg{\rvert}_{0}\right),$ (23)

where $S$ is the area of the disk. Thus, the flux of the normal component of
the angular momentum through the area is proportional to the square of its
area and to the curl of $\sigma_{ij}$, taken with respect to the first index.

If the expression in parentheses is denoted by $\Omega_{n}$

$\displaystyle\Omega_{n}=\left(\frac{\partial\sigma_{31}}{\partial
x_{2}}-\frac{\partial\sigma_{21}}{\partial x_{3}}\right)\bigg{\rvert}_{0},$
(24)

then the torque acting on the area orthogonal to the radius vector is written
as

$\displaystyle\frac{{\rm d}L_{n}}{{\rm d}t}=\frac{1}{4\pi}S^{2}\Omega_{n}.$
(25)

The non-diagonal elements of the tensor $\sigma_{ij}$ decrease with distance
as $1/r^{3}$, the curl of these components decreases as $1/r^{4}$, therefore
the torque acting on the area of finite dimensions decreases with distance as
$1/r^{4}$, as for example in Eq. (37). This is associated with difficulties in
measuring the torque of radiation at large distances.

Obviously, the calculations performed can be generalized to areas orthogonal
to other coordinate lines. The curl of the stress tensor $\sigma_{ij}$ is a
tensor of the second rank

$\displaystyle R_{ij}=e_{kli}\frac{\partial\sigma_{lj}}{\partial x_{k}}.$ (26)

From the diagonal elements of this tensor, we can form the vector

$\displaystyle\bm{\Omega}=\left(\frac{\partial\sigma_{31}}{\partial
x_{2}}-\frac{\partial\sigma_{21}}{\partial
x_{3}},\frac{\partial\sigma_{12}}{\partial
x_{3}}-\frac{\partial\sigma_{32}}{\partial
x_{1}},\frac{\partial\sigma_{23}}{\partial
x_{1}}-\frac{\partial\sigma_{13}}{\partial x_{2}}\right),$ (27)

which, by analogy with the hydrodynamics [34], can be called the vorticity of
the electromagnetic field. Sometimes this term is used to refer to the curl of
the Poynting vector of a vortex laser beam [35]. The components of the vector
$\bm{\Omega}$ designate the torque acting on the area of finite size,
orthogonal to the corresponding axis. In the experiment with a twisted laser
beam [28], the rotation of small particles away from the axis of the laser
beam was observed. This is probably a manifestation of the nonzero vorticity
of the laser beam in this direction.

Speaking of torque acting on the area, we mean the flux of the normal
component of angular momentum through the area. Actually, the angular momentum
absorbed by a real object depends significantly on the optical properties of
the material of the object and on diffraction at its edges.

Until now, we have not made any assumptions about the distance between the
charge and the point of observation. Let us now find the asymptotic
expressions for the torque of the field at distances $r,R\gg r^{\prime}$. We
represent the vectors $\bm{E}$ and $\bm{H}$ as the sum of the ‘large’ and
‘small’ components $\bm{E}=\bm{E}_{1}+\bm{E}_{2}$ and
$\bm{H}=\bm{H}_{1}+\bm{H}_{2}$. We introduce a vector $\bm{g}_{\perp}$ with
components $\sigma_{i1}$. The vector $\bm{g}_{\perp}$ is orthogonal to the
vector $\bm{n}$ and represents the shear force acting tangentially on the area
orthogonal to the vector $\bm{n}$. Arguing as in the derivation of Eq. (9), we
obtain a similar expression for the vector $\bm{g}_{\perp}$ in a spherical
coordinate system

$\displaystyle\bm{g}_{\perp}=-\frac{1}{4\pi}\left[\bm{E}_{1}(\bm{E}_{2}\,\bm{n})+H_{1}(\bm{H}_{2}\bm{n})\right],\quad
i=\theta,\varphi.$ (28)

Bearing in mind that in this approximation
$\bm{H}_{1}=(\bm{n}\times\bm{E}_{1})$ and taking into account the equality
(11), we obtain

$\bm{g}_{\perp}=\frac{1}{c}(\bm{n}\times(\bm{P}\times\bm{n})).$

One can see from the last expression that the vector $\bm{g}_{\perp}$ is
proportional to the transverse, with respect to $\bm{n}$, component of the
Poynting vector. Therefore, the lines shown in Fig. 4 can be interpreted as
the field lines of the transverse component of the Poynting vector.
Accordingly, the vorticity of radiation in the far-field approximation is

$\displaystyle\Omega_{n}=\frac{1}{c}(\bm{n}\operatorname{rot}\bm{P}).$ (29)

Eqs. (8), (12) and (29) show that in the far-field approximation, the angular
momentum flux is determined by the transverse component of the Poynting
vector, and the torque, acting on the area of finite dimensions is
proportional to the curl of the transverse component of the Poynting vector.

##  VI Rotating magnetic dipole

As a simple example, consider the field produced by a rotating magnetic
dipole. We chose a stationary field source as an example to avoid significant
complication of calculations associated with the dependence of the field on
the retarded time. The law of motion of the dipole vector $\bm{\mu}$ in the
Cartesian coordinate system ($x,y,z$) is

$\bm{\mu}=\mu(\cos\omega t,\sin\omega t,0),$ (30)

where $\mu$ and $\omega$ are the modulus and angular velocity of rotation of
the dipole. The electromagnetic field in a spherical coordinate system is as
follows

$\displaystyle E_{\theta}=$
$\displaystyle-\frac{\mu\omega}{r^{2}c}(\cos\tau-\rho\sin\tau),$
$\displaystyle E_{\varphi}=$
$\displaystyle-\frac{\mu\omega}{r^{2}c}\cos\theta(\sin\tau+\rho\cos\tau),$
$\displaystyle H_{r}=$
$\displaystyle\frac{2\mu}{r^{3}}\sin\theta(\cos\tau-\rho\sin\tau),$ (31)
$\displaystyle H_{\theta}=$
$\displaystyle-\frac{\mu}{r^{3}}\cos\theta(\cos\tau-\rho\sin\tau-\rho^{2}\cos\tau),$
$\displaystyle H_{\varphi}=$
$\displaystyle-\frac{\mu}{r^{3}}(\sin\tau+\rho\cos\tau-\rho^{2}\sin\tau),$

where

$\rho=\frac{\omega r}{c},\quad\tau=\omega t^{\prime}-\varphi,\quad
t^{\prime}=t-\frac{r}{c}.$

This field obviously has the properties of a vortex field. The wavefront given
by the equation $\tau=$ const is shown in Fig. 5.

Figure 5: The wavefront of the field defined by equations (31) has typical
singularity on the $z$ axis.

The Poynting vector has the components

$\displaystyle P_{r}=$ $\displaystyle
P_{0}[\rho^{3}(\sin^{2}\tau+\cos^{2}\theta\cos^{2}\tau)-\rho^{2}\sin
2\tau\sin^{2}\theta+\rho\cos
2\tau\sin^{2}\theta+\sin\tau\cos\tau\sin^{2}\theta],$ $\displaystyle
P_{\theta}=$ $\displaystyle P_{0}\sin\theta\cos\theta(\rho^{2}\sin
2\tau-2\rho\cos 2\tau-\sin 2\tau),$ (32) $\displaystyle P_{\varphi}=$
$\displaystyle 2P_{0}\sin\theta\left({\rho^{2}\sin^{2}\tau-\rho\sin
2\tau+\cos^{2}\tau}\right),$

where

$P_{0}=\frac{\mu^{2}\omega}{4\pi r^{5}}.$

After averaging over time we obtain

$\displaystyle\langle\bm{P}\rangle=\frac{\mu^{2}\omega^{7}}{8\pi
c^{6}\rho^{5}}\left[\rho^{3}(1+\cos^{2}\theta)\bm{e}_{r}+2(1+\rho^{2})\sin\theta\,\bm{e}_{\varphi}\right].$
(33)

The last equation shows that the Poynting vector is directed tangentially to
the surface of a cone with apex at the origin and an angular opening $\theta$.
Fig. 6 shows one of the lines of force of the Poynting vector.

Figure 6: Poynting vector force line

In the near zone, in the region of small $\rho$, the azimuthal component
$\langle P_{\varphi}\rangle$ prevails, and in the far zone prevails the radial
component $\langle P_{r}\rangle$, which determines the radiation intensity and
decreases as $1/r^{2}$. The orthogonal to $\bm{r}$ component, responsible for
the angular momentum, is relativelly large in the near field region and falls
off as $1/r^{3}$ in the far field region.

The angular momentum flux is determined by the off-diagonal elements of the
stress tensor

$\displaystyle\sigma_{12}=$ $\displaystyle\frac{\mu^{2}}{4\pi r^{6}}\sin
2\theta\left(\cos^{2}\tau-\rho\sin 2\tau-\rho^{2}\cos
2\tau+\frac{1}{2}\rho^{3}\sin 2\tau\right),$ $\displaystyle\sigma_{13}=$
$\displaystyle\frac{\mu^{2}}{4\pi r^{6}}\sin\theta(\sin 2\tau+2\rho\cos
2\tau-2\rho^{2}\sin 2\tau+2\rho^{3}\sin^{2}\tau),$ (34)
$\displaystyle\sigma_{23}=$ $\displaystyle-\frac{\mu^{2}}{4\pi
r^{6}}\cos\theta\left(\frac{1}{2}\sin 2\tau+\rho\cos 2\tau-\rho^{2}\sin
2\tau\right),$

Substituting this into the Eq. (5) and averaging over time, we obtain the
angular momentum flux in the radial direction at any distance from the dipole

$\displaystyle\left\langle\frac{{\rm d}\bm{L}}{{\rm d}t{\rm
d}s}\right\rangle=\frac{\mu^{2}\sin\theta}{4\pi
r^{5}}\left(\bm{e}_{\varphi}\cos\theta-\bm{e}_{\theta}\rho^{3}\right).$ (35)

At larger distances, only the $\theta$-component of flux remains in radiation

$\displaystyle\left\langle\frac{{\rm d}\bm{L}}{{\rm d}t{\rm
d}\Omega}\right\rangle=-\frac{\mu^{2}\omega^{3}}{4\pi
c^{3}}\sin\theta\,\bm{e}_{\theta}.$ (36)

Note, that the angular momentum flux has its maximum in direction
$\theta=\pi/2$ and is zero on the axis of rotation. Eq. (36) can be obtained
by use of Eq. (12) if the Poynting vector (33) is averaged over time and the
term with the highest power of $r$ remains. However, at arbitrary distances
Eq. (12) is not correct.

The integral of the expression (36) over the solid angle gives the well-known
formula for the rate of loss of the angular momentum of the dipole due to
radiation reaction

$\left\langle\frac{{\rm d}\bm{L}}{{\rm
d}t}\right\rangle=\frac{2\mu^{2}\omega^{3}}{3c^{3}}\bm{\hat{z}}.$

In order to find the vorticity of the electromagnetic field, we calculate the
radial component of the vector $\bm{\Omega}$. According to Eq. (24), we have

$\displaystyle\Omega_{n}=\frac{1}{r\sin\theta}\left(\frac{\partial(\sigma_{31}\sin\theta)}{\partial\theta}-\frac{\partial\sigma_{21}}{\partial\varphi}\right)=\frac{\mu^{2}\omega^{3}}{2\pi
c^{3}r^{4}}\cos\theta.$ (37)

This value, in contrast to the angular momentum flux, is maximum in the
direction of the $z$ axis, that is, in the direction of the vortex axis (see
Fig. 5). In the equatorial plane $xy$, the vorticity is zero. This property
resembles the polarization of radiation. It can be seen from Eqs. (31) that in
the $z$ axis direction the field is circularly polarized, and in the
equatorial plane it is linearly polarized, since

$\frac{E_{\varphi}^{2}}{\cos^{2}\theta}+E_{\theta}^{2}=\text{const}.$

The sign of $\Omega_{n}$ changes when passing through the equatorial plane -
the sign of vorticity coincides with the sign of the projection of the angular
velocity of rotation of the dipole onto the direction of radiation. We also
note that $\Omega_{n}$ decreases with distance as $1/r^{4}$ and this
dependence is the same at both large and small distances from the dipole.

Finally, we calculate the radial component of the curl of the Poynting vector.
Using Eqs. (VI) and averaging over time we obtain

$\langle(\bm{n}\operatorname{rot}\bm{P})\rangle=\frac{\mu^{2}\omega^{3}}{2\pi
c^{2}r^{4}}\cos\theta(1+\rho^{-2}).$

At large distance this coincides with the vorticity of radiation $\Omega_{n}$
according to Eq. (29).

##  VII Discussion

We investigated the angular momentum of an electromagnetic field of an
arbitrarily moving point charge. In the general case, the angular momentum
flux is determined by the stress tensor of the electromagnetic field (Eq.
(5)). From a practical point of view, the angular momentum flux at large
distances from the charge is of interest. Namely, at distances much greater
than the characteristic wavelength of radiation. This area is usually referred
to as a wave zone. We have shown that the flux of angular momentum in the wave
zone is proportional to the vector product of the radius vector of the
observation point by the Poynting vector (Eqs (8) and (12). Since the angular
momentum depends significantly on the choice of coordinate system, two
limiting cases can be distinguished:

i) the distance between the charge and the origin is comparable to the
distance between the charge and the point of observation, and

ii) the distance between the charge and the origin is much less than the
distance between the charge and the point of observation.

In the first case, the flux of the angular momentum of radiation can be
interpreted as the pressure of radiation on a target that produces a torque
around the origin of the coordinate system. In this case, one can take into
account only the main part of electromagnetic field decreasing with distance
as $1/r$ and assume that the vectors of the electric field, magnetic field,
and the Poynting vector are mutually orthogonal.

In the second case, it is necessary to take into account the components of the
electromagnetic field that decrease with distance as $1/r^{2}$. Thus, we take
into account that the direction of the Poynting vector does not coincide with
the direction of radiation, more precisely, with the direction of the radius
vector of the observation point. In this case, the flux of the angular
momentum of the radiation is determined by the component of the Poynting
vector that is transverse to the radius vector. In this last case, we obtained
an explicit expression for the flux of angular momentum of radiation as a
function of coordinates, velocity and acceleration of the charge (formula
(14)).

For practical applications, an important property of radiation is the torque
acting on the object due to electromagnetic field. For example, to use
radiation as optical tweeze rs and optical traps [31, 32, 27, 33]. From this
point of view, it is of interest to find the flux of the radial component of
the angular momentum through an area of finite dimensions located
perpendicular to the radius vector. We have shown that this quantity, denoted
as the vorticity of the radiation, is proportional to the curl of the stress
tensor (Eq. (27)), and in the wave zone it is proportional to the curl of the
Poynting vector (Eq. (29)).

###### Acknowledgements.

The research was supported by Ministry of Science and Higher Education of
Russian Federation, project FEWF-2020-0003; and by RFBR project No.
19-42-700011.

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# Optical modeling for the evaluation of HOWFSC on embedded processors

Kian Milani  James C. Wyant College of Optical Sciences  Ewan Douglas  Steward
Observatory  Leonid Pogorelyuk  Rensselaer Polytechnic Institute  Christopher
Mendillo  University of Massachusetts Lowell  Kerri Cahoy  Massachusetts
Institute of Technology  Nicholas Belsten  Massachusetts Institute of
Technology  Brandon Eickert  Massachusetts Institute of Technology  Shanti Rao

###### Abstract

The correction of quasi-static wavefront errors within a coronagraphic optical
system will be a key challenge to overcome in order to directly image
exoplanets in reflected light. These quasi-static errors are caused by mid to
high-order surface errors on the optical elements as a result of manufacturing
processes. Using high-order wavefront sensing and control (HOWFSC) techniques
that do not introduce non-common path aberrations, the quasi-static errors can
be corrected within the desired region of interest designated as the dark
hole. For the future Habitable Worlds Observatory (HWO), HOWFSC algorithms
will be key to attaining the desired contrasts.

To simulate the performance of HOWFSC with space rated processors, optical
models for a 6 m class space-borne observatory and a coronagraph have been
developed. Phenomena such as the Talbot effect and beamwalk are included in
the simulations using combinations of ray-based modeling and end-to-end
propagation techniques. After integrating the optical models with the embedded
processors, simulations with realistic computation times can be performed to
understand the computational hardware performance that will be needed to
maintain the desired contrasts. Here, the details of the optical models are
presented along with the HOWFSC methods utilized. Initial results of the
HOWFSC methods are also included as a demonstration of how system drifts will
degrade the contrast and require dark hole maintenance.

###### keywords:

coronagraph, dark-hole, deformable mirrors, contrast

##  1 Introduction

At the recommendation of the Astro2020 decadal survey, NASA was advised to
pursue a Habitable Worlds Observatory (HWO) equipped with a coronagraph
capable of detecting exoplanets at 1E-10 contrast levels[1]. The Nancy Grace
Roman Coronagraph Instrument will be a vital technology demonstration for the
HWO coronagraph as it will utilize high-order wavefront sensing and control
(HOWFSC) methods in order to reach contrasts on the order of 1E-8. To do so,
the Roman Coronagraph will utilize a ”set-and-forget” HOWFSC scheme involving
ground-in-the-loop operations. However, this scheme will likely be unfeasible
for the HWO coronagraph due to the higher sensitivity to drifts in the optical
system at better contrasts. This means both reaching and maintaining the 1E-10
contrast will likely require HOWFSC iterations on time scales on the order of
seconds to minutes[2]. To accomplish this, a continuous HOWFSC scheme will
have to be implemented to perform dark hole maintenance. This scheme will
require embedded processors capable of performing the HOWFSC computations
within the coherence time of the coronagraphic speckles.

Because flight hardware can be ”frozen” up to 10 years in advance of the
mission launch, work has begun on implementing these HOWFSC algorithms on
radiation hardened-processors to assess their performance, identify
computational bottlenecks, and evaluate the impact on the performance of the
instrument. To accomplish the third goal, we have implemented a framework for
modeling a potential HWO coronagraph and simulating optical disturbances such
as beamwalk. Here, the details of the optical models are presented while the
methodology and implmentation of HOWFSC algorithms on space-rated processors
is presented in Belsten et al[3]. Currently, the only HOWFSC algorithms being
investigated are pair-wise probing (PWP) and electric field conjugation
(EFC)[4].

##  2 Telescope Model

To begin with, a nominal design for an off-axis three mirror anastigmat (TMA)
telescope was created with Zemax to serve as a baseline for what a HWO could
look like. This model is loosely based on the LUVOIR-B prescription in that it
contains a similar net focal ratio of about F/36 and a similar primary mirror
focal ratio of about F/3[5]. However, the total pupil diameter has been shrunk
from the 8.4m for LUVOIR-B to 6.5m. Additionally, an unobscured circular pupil
is assumed unlike the hexagonally segmented pupil of LUVOIR-B. Note that this
telescope is not completely optimized for diffraction limited imaging over a
large FOV that will be desired for a HWO, but merely serves as a general
architecture for the optical models required. As details about the HWO become
available, this model will be updated to be more accurate.

Figure 1: This depicts the nominal design of the 6.5m class telescope
currently being used for these optical models. The telescope is assumed to use
a TMA architecture such that it can have a wider field of view to serve
multiple instruments. Here, the field traced in green is designated as the
coronagraph input as it has the smallest incident angles on the optics.

Using the raytrace model, a Fresnel model of the telescope is constructed
using POPPY as the backend propagation software[6]. This model allows for the
surface roughness of the telescope optics to be taken into account when
performing the complete optical propagation including the coronagraph. But
prior to including the WFE from each surface, the Fresnel model is validated
by comparing the footprint diameter of each optic with the calculated
footprint from the raytrace model. Additionally, the wavefront at M4 is
analyzed to ensure it is a reimaged pupil as is expected from the Zemax
design. Finally, the PSF of the Fresnel model is validated by computing the
expected resolution of the telescope with the wavelength and F-number and
comparing this resolution with the PSF result of the Fresnel model.

Figure 2: Using POPPY, a Fresnel model of the telescope is constructed to
propagate the effects from high-order surface roughness errors. Here, the
image to the left is a PSF from the Fresnel model without the WFE
contributions. The overlay of the circles at 1.22$\lambda/D$ , 5.22$\lambda/D$
, and 10.22$\lambda/D$ demonstrate that the Fresnel model is in agreement with
the expected resolution computed using the F-number from the raytrace model.
The PSF to the right is the result after including each surface’s WFE
contribution.

Using POPPY’s StatisticalPSDWFE functionality, WFE maps are pre-computed for
each optic in the telescope model. At the moment, the RMS WFE from the surface
roughness of M1-M4 is 40nm, 20nm, 20nm and 15nm respectively. The current PSD
of each surface is defined with a simple power law that has an index of -2.75,
but this PSD will be updated in the future with more realistic parameters.
These pre-computed WFEs are then used to simulate the effects of beamwalk on
M2 and M3. Given M1 and M4 are pupils, beamwalk is not considered for these
optics. Additionally, because M4 is a relayed pupil, we assume it can act as
FSM such that beamwalk from pointing errors can be neglected on downstream
optics. To implement the beamwalk, the footprint shift on M2 and M3 is assumed
to be linear with respect to pointing error, so the Zemax model is used to
compute the shift per milliarcsecond of pointing error. These values are found
to be 0.084micron/mas and 0.908micron/mas for M2 and M3 respectively. Similar
to Mendillo et al.[7], the WFE maps for each surface are shifted by the
appropriate values with subpixel precision. For this model,
scipy.ndimage.shift (or the CuPy equivalent) are used. Figure 3 illustrates
the difference in WFE for each optic for 15mas of pointing error as well as
the difference in the final pupil of the telescope computed by performing the
propagation with and without the shifted WFEs.

Figure 3: Here, the difference in WFE on both M2 and M3 are are illustrated in
the top row after computing the beamwalk induced by 15mas of pointing error.
In the bottom row are the differences in both the exit pupil’s amplitude and
OPD after computing the effect of beamwalk and comparing with the nominal exit
pupil wavefront (no pointing error).

##  3 Coronagraph Models

To perform the HOWFSC experiments with a space-rated processor in the loop,
additional Fresnel models have been created to simulate images from a ”true”
coronagraph. This Fresnel model also uses POPPY for the backend propagation
and runs on a standard PC that will be used to compute the ”true” images while
the HOWFSC algorithm running on the space-rated processor will use the images
to compute DM commands that are fed back to the coronagraph model. While
computations are performed on the processor, system drifts can be simulated in
both the telescope and coronagraph Fresnel models to evaluate the impact of
compute times. At the moment, there are no relay optics included in the design
because beamwalk or other drifts from these optics are assumed to be
negligible given they are after the FSM/M4. This allows the coroangraph model
to be slightly simplified such that the wavefront of the telescope exit pupil
is computed and directly injected into the coronagraph model rather than being
propagated through relay optics.

Here, only a simple vortex coronagraph is considered as VVCs have previously
been considered for a HabEx mission[8]. Because our telescope model does not
include any segmentation, the coronagraph model does not include any
apodization or DM assistance for the vortex, although, an additional pupil
plane where an apodizer may be placed is included so the model may be updated
in the future. Figure 4 illustrates the fundamental optical train of the
coronagraph model. Here, the deformable mirrors (DMs) are being modeled using
the fast convolution method described in Will et al.[9]. To numerically model
a vortex phase mask, the same method described in Krist et al[8] is being
used. When implementing this in a Fresnel model using POPPY, the model is
separated into two segments. The first propagates from the entrance pupil of
the coronagraph to the focal plane where the vortex will be located. At this
point, the propagation through POPPY is ended and an FFT is used to compute
the pupil plane wavefront from the focal plane data. Now, the vortex is
numerically applied with additional FFTs and MFTs which output the wavefront
at a pupil plane after the vortex. Angular spectrum propagation is then used
to back propagate the pupil wavefront to an OAP that would collimate the beam
coming from the vortex mask. A surface roughness map is applied to this
wavefront and angular spectrum is used to propagate back to the pupil. This
data now acts as the wavefront at the Lyot stop, so POPPY is again used to
propagate from the Lyot stop through the rest of the optical train and to the
image plane.

Figure 4: This figure depicts the main optical components being used for the
vortex coronagraph. The input WFE at the entrance pupil of the coronagraph is
computed using the Fresnel model of the telescope and injected into the
coronagraph model.

With the Fresnel model acting as the true coronagraph, a compact/Fraunhofer
model is also created using FFTs and MFTs for the implementation of HOWFSC
algorithms. The pre-vortex WFE is then computed with the Fresnel model and
injected into the compact model. In reality, phase retrieval techniques would
be used to measure the WFE of an instrument and inject the measurement into
the model, but no phase retrieval method has currently been implemented.
Figure 5 presents a comparison of the Fresnel model PSFs and coronagraphic
images with the injected WFE. Because the pre-FPM errors are more significant
to the coronagraph, the residual surface errors after the FPM are ignored in
this compact model. Nonetheless, the morphology of the PSF and speckles in the
compact model demonstrates agreement with the Fresnel model, so it acts as a
well calibrated model for HOWFSC.

Figure 5: Here, the images in the top demonstrate a comparison of the PSF from
the Fresnel model of the coronagraph and the PSF from the Fraunhofer model. In
the bottom row is a similar comparison with coronagraphic images of each
model. The morphology of the PSFs and coronagraphic images demonstrate the
agreement between the two models.

For most HOWFSC algorithms, the computational complexity will be dependent on
the number of actuators (or DM modes) being utilized and the number of pixels
in the focal plane within the desired control region. To evaluate the
performance of processors for varying actuator counts, two configurations of
this coronagraph model are created. The first assumes smaller 34x34 actuator
DMs while the second uses 68x68 actuator DMs. The larger DM model assumes the
same actuator spacing, but the pupil diameter is doubled to account for the
higher actuator count. The detector sampling in each model is assumed to be 5
microns, but the final imaging focal length is also doubled for the model with
larger pupils such that the pixelscale is 0.354$\lambda/D$ in each. Table 1
contains the details of each models pupil sizes and actuator counts while
Figure 6 illustrates the difference in potential control regions with the
higher actuator count. Both models will be used when evaluating the processors
to understand how the performance will scale with more actuators and larger
control regions.

Table 1: The assumptions for the two DM configurations being considered for the HOWFSC implementation are presented here. |  Small DM model |  Large DM model
---|---|---
Actuators across DM |  34 |  68
Total Actuator Count |  $2\times 952$ |  $2\times 3720$
Actuator Spacing |  0.3mm |  0.3mm
Pupil Size at DM |  9.6mm |  19.2mm
Actuators Across Pupil |  32.0 |  64.0

Figure 6: The images here demonstrate the control regions for each DM
configuration being considered using a waffle command to produce speckles at
the edges of each potential control region. Note that here, $D$ is the
diameter of Lyot stop, which is undersized from the geometric pupil diameter.

##  4 HOWFSC Simulations

Currently, the primary HOWFSC methods being considered for dark hole creation
and maintenance are standard EFC and PWP. Two additional HOWFSC methods will
be implemented in the future including modal PWP demonstrated by Pogorelyuk et
al.[10] for more efficient dark hole maintenance along with the Jacobian-free
algorithmic differentiation EFC introduced by Will et al[11]. For now, the
models discussed above are used to perform EFC and illustrate why dark hole
maintenance will be necessary, particularly for the 1E-10 contrasts.

For simplicity, only monochromatic EFC is implemented, but this will be
expanded to broadband EFC in the future. Figures 7 and 8 illustrate an example
of why using a HOWFSC method for dark hole maintenance will likely be
necessary. Here, EFC is originally used to create a dark hole assuming a
perfectly stable telescope. This allows us to generate a dark hole with 1E-10
contrast in just 18 iterations. One noteworthy point is that due to the large
WFEs assumed in the telescope optics, the Jacobian was relinearized after 9
iterations in order to reach the final contrasts indicated in the figures.
After the initial EFC loop is completed, a pointing error of 15mas was
injected into the telescope model. The effect of beamwalk on M2 and M3 was
then computed and injected into the coronagraph model to assess the contrast
degradation this level of pointing error would induce assuming no other
dynamics in the coronagraph system. As a result of this beamwalk, the contrast
degrades by about an order magnitude with each DM configuration. However, the
new speckles can be corrected with additional iterations of EFC. For the
models being used here, each DM configuration utilized 3 more EFC iteration to
re-converge to 1E-10 contrast. It should be noted that contrast degradation
from beamwalk and other drifts in the coronagraph system will also depend on
the quality of each optic’s surface roughness as the better each optic can be
polished, then larger drifts can be tolerated.

Figure 7: Using the Fraunhofer model of for each DM configuration of the
coronagraph, EFC is used to produce a dark hole. In the top row are the
solutions for the 34x34 DMs and the bottom row are the solutions with the
68x68 DMs. Each of the solutions are generated after 18 iterations of EFC
using a single relinearization after 9 iterations.

Figure 8: In the top row are the images from each coronagraph model if we
assume the telescope pointing drifts by 15mas (x + y). Here, the contrast in
each dark hole degrades by about 1E-9 solely from the beamwalk on M2 and M3.
In the bottom row, 3 additional iterations of EFC are used to correct the
drift in the speckles and re-obtain the 1E-10 contrast with each DM
configuration.

##  5 Conclusions and Future Work

While a set-and-forget HOWFSC scheme is possible for more moderate contrast
goals, the extreme contrast requirement of the HWO will require much more
frequent dark hole maintenance with HOWFSC techniques due to drifts within the
optical system. These drifts will have to be corrected on time scales
equivalent to or smaller than the lifetime of the speckles in order for the
HOWFSC maintenance to operate as intended. Here a framework has been developed
to model various configurations and parameters of a 6.5m class telescope and
coronagraph to evaluate the impact of HOWFSC computation times. As more
details become available about potential HWO concepts including telescope
design, coronagraph modes, and quality of optical surfaces, the models here
will be updated to include the more accurate parameters. Future experiments
with space-rated processors in-the-loop will also implement various other
dynamics such as DM creep and slow shifts of coronagraph optics. These
experiments will yield results informing the mission about how fast HOWFSC
will need to run for various conditions and what the computational bottlenecks
for HOWFSC will be.

##  6 Acknowledgements

This work is supported by the NASA Astrophysics Technology Division under APRA
grant #80NSSC22K1412. This research made use of community-developed core
Python packages, including: POPPY[6], Astropy [12], Matplotlib [13], SciPy
[14], CuPy[15], Ray[16], and the IPython Interactive Computing architecture
[17].

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