Radiation of a Massless Closed String from Interacting D$p$-branes with
Background Fields in the Superstring Theory

Hamidreza Daniali and Davoud Kamani

Department of Physics, Amirkabir University of Technology (Tehran
Polytechnic), Iran
P.O.Box: 15875-4413
e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

In the context of the superstring theory, we calculate the radiation amplitude
of a general massless closed string from the interaction of two parallel
D$p$-branes. The background field is the constant antisymmetric tensor
$B_{\mu\nu}$, and the branes have been equipped with the different $U(1)$
gauge fields. For the large distance of the branes, the Kalb-Ramond emission
will be investigated.

PACS numbers: 11.25.-w; 11.25.Uv

Keywords: Boundary state; Background fields; Radiation amplitude; The Kalb-
Ramond state.

##  1 Introduction

The study of the D-branes with nonzero background and internal gauge fields,
specially their interactions, reveals a number of interesting properties in
the string theory [1]-[13]. In the closed string channel, the interaction
between the branes is realized as the emission/absorption of a closed string
between the branes. A practical and adequate technique for calculating such
interactions is the boundary state formalism [2], [4], [8], [14]-[20]. This
method allows us to explore some of the branes properties which are not
accessible via the other formalisms.

Besides the branes interactions, the closed string radiation from the branes
has been also studied. One of the most remarkable setups is the closed string
production from a single unstable brane [21]-[24]. This sort of radiation has
been investigated in various configurations [25]-[29]. The closed string
radiation from the interacting branes is another appealing configuration
[30]-[33]. In this formalism, an appropriate vertex operator is sandwiched
between the boundary states, which correspond to the branes.

In this paper, we shall concentrate on the closed string radiation from the
interacting parallel D$p$-branes. Our branes have been equipped with the Kalb-
Ramond field and the internal $U(1)$ gauge potentials. For calculating the
amplitude of the radiation, we shall use the boundary state formalism. In the
large distance of the branes and in the low energy limit, the massless closed
strings can be usually emitted in the three distinct physical processes: the
Bremsstrahlung-like process, which denotes the emission from each of the
interacting branes, and an emission from the middle region between the branes
[31, 33]. These results drastically encouraged and motivated us to investigate
the supersymmetric version of the closed string radiation from the interacting
D$p$-branes.

This paper is organized as follows. In Sec. 2, the boundary state,
corresponding to a D$p$-brane with the internal and background fields, will be
reviewed. In Sec. 3, we shall calculate the radiation amplitude of a general
massless closed string from the interaction of two parallel dressed
D$p$-branes. In Sec. 4, the amplitude for the Kalb-Ramond state radiation from
the branes with large distance will be presented. Our conclusions have been
given in Sec. 5.

##  2 The boundary state

For constructing the boundary state, which corresponds to a D$p$-brane with
the background field and internal gauge potential, we start with the well-
known closed string action

$S_{\sigma}=-\dfrac{1}{4\pi\alpha^{\prime}}\int_{\Sigma}d^{2}\sigma\left(\sqrt{-h}h^{ab}G_{\mu\nu}+\epsilon^{ab}B_{\mu\nu}\right)\partial_{a}X^{\mu}\partial_{b}X^{\nu}+\dfrac{1}{2\pi\alpha^{\prime}}\int_{\partial\Sigma}d\sigma
A_{\alpha}\partial_{\sigma}X^{\alpha},$ (2.1)

where $\mu,\nu$ are the spacetime indices, i.e.,
$\mu,\nu\in\\{0,1,\cdots,9\\}$, and $\alpha,\beta\in\\{0,1,\cdots,p\\}$
represent the worldvolume directions of the D$p$-brane. For the perpendicular
directions to the worldvolume, we shall apply $i,j\in\\{p+1,\cdots,9\\}$. We
use the flat spacetime with the metric $G_{\mu\nu}={\rm diag}(-1,1,\cdots,1)$,
flat worldsheet $h_{ab}=\eta_{ab}$ with $a,b\in\\{\tau,\sigma\\}$, and a
constant Kalb-Ramond field $B_{\mu\nu}$. Besides, we adopt the Landau gauge
$A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}X^{\beta}$ with the constant field
strength $F_{\alpha\beta}$.

The equation of motion and the boundary state equations are derived by
vanishing the variation of the action. Consequently, using the coherent state
approach [34] and the closed string mode expansion, the oscillating part of
the bosonic portion of the boundary state takes the form

$\displaystyle|B_{x}\rangle_{\rm osc}$ $\displaystyle=$
$\displaystyle\sqrt{-\det(\mathbf{1}-\mathcal{F})}\exp\left[-\sum_{m=1}^{\infty}\left(\dfrac{1}{m}\alpha^{\mu}_{-m}{S}_{\mu\nu}{\tilde{\alpha}}^{\nu}_{-m}\right)\right]|0\rangle,$
(2.2)

in which, we defined

${S}^{\mu}_{\ \nu}=\left\\{Q^{\alpha}_{\
\beta}\equiv\left(\dfrac{\mathbf{1}+\mathcal{F}}{\mathbf{1}-\mathcal{F}}\right)^{\alpha}_{\;\;\;\beta},-\delta^{i}_{\;j}\right\\},\qquad\mathcal{F}_{\alpha\beta}\equiv
F_{\alpha\beta}-B_{\alpha\beta}.$ (2.3)

The normalization factor $\sqrt{-\det(\mathbf{1}-\mathcal{F})}$ comes from the
disk partition function [7]. The zero-mode part of the boundary state
possesses the feature

$|B_{x}\rangle^{(0)}=\int_{-\infty}^{+\infty}\prod_{i=p+1}^{9}\left[\dfrac{dP^{i}}{2\pi}\exp\left(-iP^{i}y^{i}\right)|P^{i}\rangle\right]\prod_{\alpha}|P^{\alpha}=0\rangle.$
(2.4)

The momentum $P^{\mu}$ has nonzero components only in the transverse
directions.

For receiving the boundary state equations of the worldsheet fermions, we
should apply the supersymmetry transformations on the bosonic boundary state
equations. Accordingly, the following substitutions should be exerted in the
bosonic boundary state equations

$\displaystyle\partial_{+}X^{\mu}(\sigma,\tau)\rightarrow-i\eta\psi^{\mu}_{+}(\sigma,\tau),\qquad\partial_{-}X^{\mu}(\sigma,\tau)\rightarrow\psi^{\mu}_{-}(\sigma,\tau),$
(2.5)

where $\partial_{\pm}\equiv\frac{1}{2}(\partial_{\tau}\pm\partial_{\sigma})$,
and $\eta=\pm 1$ will be used for applying the GSO projection on the boundary
states.

Let us denote the fermionic oscillators in the R-R (NS-NS) sector with
$d_{m}^{\mu}$ ($b_{r}^{\mu}$). Thus, we acquire

$|B_{\psi},\eta\rangle_{\rm
NS}=\exp\left(i\eta\sum_{r=1/2}^{\infty}b^{\mu}_{-r}S_{\mu\nu}\tilde{b}_{-r}^{\nu}\right)|0\rangle_{\rm
NS},$ (2.6)

which represents the fermionic part of the NS-NS sector boundary state, and

$\displaystyle|B_{\psi},\eta\rangle_{\rm
R}=\dfrac{1}{\sqrt{-\det(\mathbf{1}-\mathcal{F}})}\exp\left(i\eta\sum_{m=1}^{\infty}d^{\mu}_{-m}S_{\mu\nu}\tilde{d}^{\nu}_{-m}\right)|B_{\psi},\eta\rangle_{\rm
R}^{(0)},$ (2.7)

is the fermionic part of the R-R sector boundary state. For both type IIA and
type IIB theories, the explicit form of the zero-mode portion of the boundary
state is given by

$|B_{\psi},\eta\rangle_{\rm
R}^{(0)}=\left(C\Gamma^{0}\cdots\Gamma^{p}\;\dfrac{1+i\eta\Gamma^{11}}{1+i\eta}\mathcal{G}\right)_{AB}|A\rangle\otimes|\tilde{B}\rangle,$
(2.8)

where $A$ and $B$ denote the 32-dimensional indices for the $\Gamma$-matrices
and spinors in the 10-dimensional spacetime,
$|A\rangle\otimes|\tilde{B}\rangle$ is the vacuum of the zero modes
$d_{0}^{\mu}$ and $\tilde{d}_{0}^{\mu}$, and $C$ is the charge conjugate
matrix. Furthermore, the matrix $\mathcal{G}$ must satisfy the equation
$\Gamma^{\alpha}\mathcal{G}=Q^{\alpha}_{\ \beta}\;\mathcal{G}\Gamma^{\beta}$.
Hence, this matrix possesses the following conventional solution

$\mathcal{G}=*\exp\left(\frac{1}{2}\mathcal{F}_{\alpha\beta}\Gamma^{\alpha}\Gamma^{\beta}\right)*.$
(2.9)

The notation $*\ *$ indicates that the exponential should be expanded such
that all $\Gamma$-matrices anticommute. Thus, we have a finite number of
terms.

The following direct product exhibits the total boundary state, associated
with the D$p$-brane, for both the NS-NS and R-R sectors

$|B,\eta\rangle_{\rm NS(R)}=\frac{T_{p}}{2}|B_{x}\rangle\otimes|B^{\rm
gh}\rangle\otimes|B_{\psi},\eta\rangle_{\rm NS(R)}\otimes|B^{\rm
sgh},\eta\rangle_{\rm NS(R)},$ (2.10)

where $T_{p}$ is the brane tension. Note that the background and internal
fields have no effect on the ghost and superghost parts of the boundary
states. Since the explicit expressions of $|B^{\rm gh}\rangle$ and $|B^{\rm
sgh},\eta\rangle_{\rm NS(R)}$ are available in the literature, we shall not
write them here.

##  3 The closed string radiation

In the closed string channel, two D-branes interact by exchanging a closed
string between two boundary states, associated with the D-branes. Let $\sigma$
be the periodic coordinate of the string worldsheet, i.e.
$0\leq\sigma\leq\pi$, and $0\leq\tau\leq t$ be its length coordinate. Besides,
the fields on the two D$p$-branes are labeled with the subscripts 1 and 2. In
addition, we shall apply the momenta $k_{1}$ and $k_{2}$ for the emitted
closed string from the first brane and the absorbed one by the second brane,
respectively. The radiation of a closed string state via the interacting
branes is elaborated by inserting the associated vertex operator of the string
state, $V(\sigma,\tau)$, into the interaction amplitude

$\mathcal{A}_{\rm NS(R)}(\eta_{1},\eta_{2})=\int_{0}^{\infty}{\rm
d}t\int_{0}^{t}{\rm d}\tau\int_{0}^{\pi}{\rm d}\sigma\;_{\rm NS(R)}\langle
B^{1},\eta_{1}|e^{-tH_{\rm NS(R)}}V(\sigma,\tau)|B^{2},\eta_{2}\rangle_{\rm
NS(R)},$ (3.1)

where $H_{\rm NS}$ ($H_{\rm R}$) represents the total closed string
Hamiltonian in the NS-NS (R-R) sector. Note that, because of the effects of
the GSO projection, the total amplitude is given by the summation over the
spin structures.

Let us define $z\equiv\sigma+i\tau$ and $\partial\equiv\partial_{z}$. The
vertex operator in Eq. (3.1) for a general massless closed string has the
following form

$\displaystyle V(z,\bar{z})=\epsilon_{\mu\nu}\left(\partial
X^{\mu}(z,\bar{z})-\dfrac{1}{2}p\cdot\psi(z)\psi^{\mu}(z)\right)\left(\bar{\partial}X^{\nu}(z,\bar{z})+\dfrac{1}{2}p\cdot\tilde{\psi}(\bar{z})\tilde{\psi}^{\nu}(\bar{z})\right)e^{ip\cdot
X(z,\bar{z})},$ (3.2)

in which $\epsilon_{\mu\nu}$ denotes the polarization tensor, and $p$ (with
$p^{2}=0$) indicates the momentum of the radiated massless closed string. For
simplicity, similar to the Refs. [31], [35] and [36], we assume that the
polarization tensor has non-vanishing elements only in the perpendicular
directions of the brane worldvolume, i.e.,
$\epsilon_{\alpha\beta}=\epsilon_{\alpha i}=\epsilon_{i\alpha}=0$.

After heavy calculations, the radiation amplitude finds the following feature

$\displaystyle\mathcal{A}_{\rm NS(R)}(\eta_{1},\eta_{2})$ $\displaystyle=$
$\displaystyle\epsilon_{ij}\left[\prod_{\alpha=0}^{p}\delta(p^{\alpha})\right]\int_{0}^{\infty}{\rm
d}t^{\prime}\int_{0}^{\infty}{\rm d}\tau\int_{0}^{\pi}{\rm
d}\sigma\int_{-\infty}^{+\infty}\prod_{i=p+1}^{9}\frac{{\rm
d}k_{1}^{i}}{2\pi}e^{(t^{\prime}+\tau)\xi}e^{ik_{1}^{i}b^{i}}$ (3.3)
$\displaystyle\times$ $\displaystyle
e^{-t^{\prime}\alpha^{\prime}k_{1}^{2}}e^{-\alpha^{\prime}\tau k_{2}^{2}}\
\mathcal{Z}^{x}\ \mathcal{Z}^{\psi}_{\rm NS(R)}(\eta_{1},\eta_{2})\ \langle
e^{ip\cdot x}\rangle\ \mathcal{M}^{ij}_{\rm NS(R)},$

where $\xi=0\ (-5/2)$ has been obtained by regularizing the Hamiltonian in the
R-R (NS-NS) sector. Using the notation

$\displaystyle\langle\mathcal{O}(\sigma,\tau)\rangle_{\rm
osc}\equiv\dfrac{{}_{\rm osc}\langle B^{1},\eta_{1}|e^{-tH_{\rm
osc}}\mathcal{O}(\sigma,\tau)|B^{2},\eta_{2}\rangle_{\rm osc}}{{}_{\rm
osc}\langle B^{1},\eta_{1}|e^{-tH_{\rm osc}}|B^{2},\eta_{2}\rangle_{\rm
osc}},$

the matrix elements $\mathcal{M}^{ij}_{\rm NS(R)}$ take the form

$\displaystyle\mathcal{M}^{ij}_{\rm NS(R)}$ $\displaystyle=$
$\displaystyle\langle\partial X^{i}\bar{\partial}X^{j}\rangle_{\rm
osc}-\langle\partial X^{i}p.X\rangle_{\rm osc}\langle\bar{\partial}X^{j}p\cdot
X\rangle_{\rm osc}-\alpha^{\prime 2}k_{1}^{i}k_{1}^{j}$ (3.4) $\displaystyle+$
$\displaystyle\frac{1}{4}\Big{(}\langle p\cdot\psi\
p\cdot\tilde{\psi}\rangle\langle\psi^{i}\tilde{\psi}^{j}\rangle-\langle
p\cdot\psi\ \psi^{i}\rangle\langle p\cdot\tilde{\psi}\
\tilde{\psi}^{j}\rangle+\langle p\cdot\tilde{\psi}\ \psi^{i}\rangle\langle
p\cdot\psi\ \tilde{\psi}^{j}\rangle\Big{)}_{\rm NS(R)}$ $\displaystyle+$
$\displaystyle\frac{i}{2}\Big{(}\langle\partial X^{i}p\cdot X\rangle_{\rm
osc}\langle p\cdot\tilde{\psi}\ \tilde{\psi}^{j}\rangle_{\rm
NS(R)}-\langle\partial X^{j}p\cdot X\rangle_{\rm osc}\langle p\cdot\psi\
\psi^{i}\rangle_{\rm NS(R)}\Big{)}$ $\displaystyle-$
$\displaystyle\alpha^{\prime}k^{i}\Big{(}i\langle\bar{\partial}X^{j}p\cdot
X\rangle_{\rm osc}+\frac{1}{2}\langle p\cdot\tilde{\psi}\
\tilde{\psi}^{j}\rangle_{\rm NS(R)}\Big{)}$ $\displaystyle+$
$\displaystyle\alpha^{\prime}k^{j}\Big{(}i\langle\partial X^{i}p\cdot
X\rangle_{\rm osc}-\frac{1}{2}\langle p\cdot\psi\ \psi^{i}\rangle_{\rm
NS(R)}\Big{)}.$

The partition functions $\mathcal{Z}^{x}$ and $\mathcal{Z}^{\psi}_{\rm
NS(R)}(\eta_{1},\eta_{2})$ are specified by

$\displaystyle\mathcal{Z}^{x}$ $\displaystyle=$ $\ {}_{\rm osc}\langle
B^{1}_{x}|e^{-tH_{\rm osc}}|B^{2}_{x}\rangle_{\rm osc}\;\langle B^{\rm
gh}|e^{-tH_{\rm gh}}|B^{\rm gh}\rangle,$ (3.5)
$\displaystyle\mathcal{Z}^{\psi}_{\rm NS(R)}(\eta_{1},\eta_{2})$
$\displaystyle=$ $\displaystyle\Bigg{(}\ _{\rm osc}\langle
B^{1}_{\psi},\eta_{1}|e^{-tH_{\rm osc}}|B^{2}_{\psi},\eta_{2}\rangle_{\rm
osc}\;\langle B^{\rm sgh},\eta_{1}|e^{-tH_{\rm sgh}}|B^{\rm
sgh},\eta_{2}\rangle\Bigg{)}_{\rm NS(R)}.$ (3.6)

In the amplitude (3.4), we defined the impact parameter
$b^{i}=y^{i}_{1}-y^{i}_{2}$. Besides, by using the Wick’s rotation
$\tau\to-i\tau$, we introduced another proper time $t^{\prime}=t-\tau$. Note
that, the presence of the factor $\prod_{\alpha=0}^{p}\delta(p^{\alpha})$
manifestly elaborates that the radiated closed string moves perpendicular to
the brane.

###  3.1 The correlators and partition functions

Given the explicit form of the Hamiltonian and the boundary states (2.2) and
(2.4), the bosonic correlators and partition functions take the features

$\mathcal{Z}^{x}=\sqrt{\det\Big{[}(\mathbf{1}-\mathcal{F}_{1})(\mathbf{1}-\mathcal{F}_{2})\Big{]}}\prod_{n=1}^{\infty}\bigg{\\{}\left[{\rm
det}\left(\mathbf{1}-{\mathcal{S}}\
q^{2n}\right)\right]^{-1}\left({\mathbf{1}}-q^{2n}\right)^{2}\bigg{\\}},$
(3.7) $\displaystyle\langle\partial X^{\mu}X^{\nu}\rangle_{\rm osc}$
$\displaystyle=$ $\displaystyle
i\alpha^{\prime}\Bigg{\\{}\eta^{\mu\nu}\sum_{n=1}^{\infty}\text{Tr}\left(\dfrac{{\mathbf{1}}+{\mathcal{S}}q^{2n}}{{\mathbf{1}}-{\mathcal{S}}q^{2n}}\right)-\sum_{n=1}^{\infty}\left(S_{2}S_{1}\right)^{\nu\mu}\text{Tr}\left(\dfrac{{\mathcal{S}}q^{2n}}{{\mathbf{1}}-{\mathcal{S}}q^{2n}}\right)$
(3.8) $\displaystyle-$
$\displaystyle\sum_{n=0}^{\infty}S^{\mu\nu}_{2}\text{Tr}\Bigg{[}\dfrac{{\mathcal{S}}q^{2n}e^{-4t^{\prime}}}{{\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4t^{\prime}}}\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4t^{\prime}}}\right)\Bigg{]}$
$\displaystyle+$
$\displaystyle\sum_{n=0}^{\infty}\left(S_{1}\right)^{\nu\mu}\text{Tr}\Bigg{[}\dfrac{{\mathcal{S}}q^{2n}e^{-4\tau}}{{\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4\tau}}\left({\mathbf{1}}+\dfrac{1}{{\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4\tau}}\right)\Bigg{]}\Bigg{\\}},$
$\displaystyle\langle e^{ip\cdot X_{\rm osc}}\rangle$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}\det\left({\mathbf{1}}-{\mathcal{S}}q^{2n}\right)^{\frac{\alpha^{\prime}}{2n}p_{\mu}p_{\nu}\left(S_{2}S_{1}\right)^{\mu\nu}}$
(3.9) $\displaystyle\times$
$\displaystyle\prod_{n=0}^{\infty}\det\left[\left({\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4t^{\prime}}\right)^{p_{\mu}p_{\nu}S^{\mu\nu}_{2}}\left({\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4\tau}\right)^{p_{\mu}p_{\nu}S_{1}^{\mu\nu}}\right]^{-\frac{\alpha^{\prime}}{2(n+1)}}$
$\displaystyle\times$
$\displaystyle\prod_{n=0}^{\infty}\det\left\\{\exp\left[\frac{\alpha^{\prime}p_{\mu}p_{\nu}S^{\mu\nu}_{2}}{2(n+1)}\left({\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4t^{\prime}}\right)^{-1}\right]\right\\}\
\ \ $ $\displaystyle\times$
$\displaystyle\prod_{n=0}^{\infty}\det\left\\{\exp\left[\frac{\alpha^{\prime}p_{\mu}p_{\nu}S_{1}^{\mu\nu}}{2(n+1)}\left({\mathbf{1}}-{\mathcal{S}}q^{2n}e^{-4\tau}\right)^{-1}\right]\right\\},$

where we defined $\mathcal{S}=S_{1}^{\rm T}S_{2}$ and $q=e^{-2t}$. Note that
$\langle{\bar{\partial}}X^{\mu}X^{\nu}\rangle_{\rm
osc}=-\langle{\partial}X^{\mu}X^{\nu}\rangle_{\rm osc}$. Using the equation of
motion $\partial\bar{\partial}X=0$, the two-derivative correlator can be
obtained from Eq. (3.8), by taking the derivative with respect to $\bar{z}$.

Now consider the fermionic section. For the NS-NS sector, by employing the
boundary state (2.6), we receive

$\displaystyle\mathcal{Z}^{\psi}_{\rm
NS}(\eta_{1},\eta_{2})=\prod_{n=1}^{\infty}\bigg{\\{}\left[\det\left(\mathbf{1}+\eta_{1}\eta_{2}\
\mathcal{S}q^{2n-1}\right)\right]\left(1+\eta_{1}\eta_{2}\
q^{2n-1}\right)^{-2}\bigg{\\}},$ (3.10)
$\displaystyle\langle\psi^{\mu}\psi^{\nu}\rangle_{\rm
NS}=\langle\tilde{\psi}^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm NS}$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\left\\{\eta^{\mu\nu}{\rm
Tr}\left(\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}}\right)\right.$ (3.11)
$\displaystyle-$
$\displaystyle\left.(-\eta_{1}\eta_{2})^{n}(S_{2}S_{1})^{\nu\mu}{\rm
Tr}\left(\frac{\mathcal{S}q^{n}}{\mathbf{1}-\mathcal{S}q^{2n}}\right)\right\\},$
$\displaystyle\langle\psi^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm NS}$
$\displaystyle=$ $\displaystyle
i\eta_{1}\sum_{n=0}^{\infty}(-\eta_{1}\eta_{2})^{n}(S_{1})^{\nu\mu}{\rm
Tr}\left[\frac{\mathcal{S}q^{n}e^{-2\tau}}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4\tau}}\left(\mathbf{1}+\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4\tau}}\right)\right]$
(3.12) $\displaystyle+$ $\displaystyle
i\eta_{2}\sum_{n=0}^{\infty}(-\eta_{1}\eta_{2})^{n}S_{2}^{\mu\nu}{\rm
Tr}\left[\frac{\mathcal{S}q^{n}e^{-2t^{\prime}}}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4t^{\prime}}}\left(\mathbf{1}+\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4t^{\prime}}}\right)\right].$

The correlator $\langle\tilde{\psi}^{\mu}\psi^{\nu}\rangle_{\rm NS}$ can be
obtained from $\langle\psi^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm NS}$ via the
replacements $S_{1}\to S^{\rm T}_{1}$ and $S_{2}\to S^{\rm T}_{2}$.

Because of the zero-mode part of the R-R sector, the computations in this
sector require specific treatment. Using Eqs. (2.7) and (2.8), the partition
function finds the form

$\displaystyle\mathcal{Z}_{R}^{\psi}(\eta_{1},\eta_{2})$ $\displaystyle=$
$\displaystyle\frac{(-1)^{p}}{\sqrt{\det\Big{[}(\mathbf{1}-\mathcal{F}_{1})(\mathbf{1}-\mathcal{F}_{2})\Big{]}}}\prod_{n=1}^{\infty}\bigg{\\{}\left[\det\left(\mathbf{1}+\eta_{1}\eta_{2}\
\mathcal{S}q^{2n}\right)\right]\left(1+\eta_{1}\eta_{2}\
q^{2n}\right)^{-2}\bigg{\\}}$ (3.13) $\displaystyle\times$
$\displaystyle\left(i\mathbf{T}^{\prime}\delta_{\eta_{1}\eta_{2},1}-\frac{1}{2}\mathbf{T}\delta_{\eta_{1}\eta_{2},-1}\right),$

in which $\mathbf{T}$ and $\mathbf{T^{\prime}}$ are defined by

$\displaystyle\mathbf{T}\equiv{\rm
Tr}\left(\mathcal{G}_{1}C^{-1}\mathcal{G}_{2}^{T}C\right),\qquad\mathbf{T}^{\prime}\equiv{\rm
Tr}\left(\mathcal{G}_{1}C^{-1}\mathcal{G}_{2}^{T}C\Gamma_{11}\right).$ (3.14)

The correlators in the R-R sector are separated into the oscillatory part and
zero-mode portion, i.e, $\langle\mathcal{O}\rangle_{\rm
R}=\langle\mathcal{O}\rangle_{\rm R}^{\rm osc}+\langle\mathcal{O}\rangle_{\rm
R}^{(0)}$. For the oscillatory part, one finds

$\displaystyle\langle\psi^{\mu}\psi^{\nu}\rangle_{\rm R}^{\rm
osc}=\langle\tilde{\psi}^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm R}^{\rm osc}$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\left\\{\eta^{\mu\nu}{\rm
Tr}\left(\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}}\right)\right.$ (3.15)
$\displaystyle-$
$\displaystyle\left.(-\eta_{1}\eta_{2})^{n}(S_{2}S_{1})^{\nu\mu}{\rm
Tr}\left(\frac{\mathcal{S}q^{2n}}{\mathbf{1}-\mathcal{S}q^{2n}}\right)\right\\},$
$\displaystyle\langle\psi^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm R}^{\rm osc}$
$\displaystyle=$ $\displaystyle
i\eta_{1}\sum_{n=0}^{\infty}(-\eta_{1}\eta_{2})^{n}(S_{1})^{\nu\mu}{\rm
Tr}\left[\frac{\mathcal{S}q^{2n}e^{-4\tau}}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4\tau}}\left(\mathbf{1}+\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4\tau}}\right)\right]$
(3.16) $\displaystyle+$ $\displaystyle
i\eta_{2}\sum_{n=0}^{\infty}(-\eta_{1}\eta_{2})^{n}S_{2}^{\mu\nu}{\rm
Tr}\left[\frac{\mathcal{S}q^{2n}e^{-4t^{\prime}}}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4t^{\prime}}}\left(\mathbf{1}+\frac{1}{\mathbf{1}-\mathcal{S}q^{2n}e^{-4t^{\prime}}}\right)\right].$

The correlator $\langle\tilde{\psi}^{\mu}\psi^{\nu}\rangle_{\rm R}$ can be
obviously obtained from $\langle\psi^{\mu}\tilde{\psi}^{\nu}\rangle_{\rm R}$
via the replacements $S_{1}\to S^{\rm T}_{1}$ and $S_{2}\to S^{\rm T}_{2}$.
For the zero-mode part, we obtain

$\displaystyle\langle d_{0}^{\mu}d_{0}^{\nu}\rangle_{\rm R}^{(0)}$
$\displaystyle=$
$\displaystyle(-1)^{p+1}\langle\tilde{d}_{0}^{\mu}\tilde{d}_{0}^{\nu}\rangle_{\rm
R}^{(0)}=\frac{(-1)^{p}}{2}\left(i\delta_{\eta_{1}\eta_{2},1}\frac{\mathcal{T}^{\prime\
\mu\nu}}{\mathbf{T}^{\prime}}-\delta_{\eta_{1}\eta_{2},-1}\frac{\mathcal{T}^{\mu\nu}}{\mathbf{T}}\right),$
(3.17) $\displaystyle\langle\tilde{d}_{0}^{\mu}d_{0}^{\nu}\rangle_{\rm
R}^{(0)}$ $\displaystyle=$ $\displaystyle-\langle
d_{0}^{\mu}\tilde{d}_{0}^{\nu}\rangle_{\rm
R}^{(0)}=\frac{(-1)^{p}}{2}\left(i\delta_{\eta_{1}\eta_{2},1}\frac{\mathcal{T}^{\mu\nu}}{\mathbf{T}^{\prime}}-\delta_{\eta_{1}\eta_{2},-1}\frac{\mathcal{T}^{\prime\
\mu\nu}}{\mathbf{T}}\right),$ (3.18)

where the new tensors are defined as follows

$\displaystyle\mathcal{T}^{\mu\nu}$ $\displaystyle\equiv$ $\displaystyle{\rm
Tr}\left(\mathcal{G}_{1}\Gamma^{\nu}C^{-1}\mathcal{G}_{2}^{T}C\Gamma^{p}\cdots\Gamma^{1}\Gamma^{\mu}\Gamma^{1}\cdots\Gamma^{p}\right),$
$\displaystyle\mathcal{T}^{\prime\ \mu\nu}$ $\displaystyle\equiv$
$\displaystyle{\rm
Tr}\left(\mathcal{G}_{1}\Gamma^{\nu}C^{-1}\mathcal{G}_{2}^{T}C\Gamma^{p}\cdots\Gamma^{1}\Gamma^{\mu}\Gamma^{1}\cdots\Gamma^{p}\Gamma_{11}\right).$
(3.19)

According to Eqs. (3.7)-(3.1), the integrand of Eq. (3.3) merely depends on
$\tau$ and $t^{\prime}$, and it is independent of the variable $\sigma$.
Hence, the integration over the worldsheet coordinate $\sigma$ is given by an
overall factor $\pi$.

By introducing all correlators and partition functions into Eq. (3.3), we
receive the finalized form of the partial amplitudes $\mathcal{A}_{\rm
NS(R)}(\eta_{1},\eta_{2})$. The total amplitude is given by

$\displaystyle\mathcal{A}_{\rm
total}=\mathcal{A}(+,+)+\mathcal{A}(-,-)+\mathcal{A}(+,-)+\mathcal{A}(-,+),$
(3.20) $\displaystyle\mathcal{A}(\eta_{1},\eta_{2})=\mathcal{A}_{\rm
NS}(\eta_{1},\eta_{2})+\mathcal{A}_{\rm R}(\eta_{1},\eta_{2}).$

Due to the very long length of the total amplitude, we do not explicitly write
it.

##  4 The emission of the Kalb-Ramond state from large distance D$p$-branes

In the case of large distance branes, which corresponds to a long enough time,
the exchange of the massless states extremely possesses the dominant
contribution to the interaction. However, in the interaction amplitude the
limit $t\to\infty$ should merely exert on the contribution of the oscillators,
i.e, on the correlators and partition functions. Besides, we should mention
that the exponent of the modular parameter $q$ in the R-R sector is not in the
same order as in the NS-NS sector (see Eqs. (3.12) and (3.16)). Therefore, in
both sectors we shall apply the order $\mathcal{O}(q^{2n})$.

In fact, even in the eikonal approximation [31, 32], in which the recoils of
the branes are ignored, the particle creation quantum-mechanically is
permitted. This fact becomes more obvious when one considers the emission of
the massless strings in the limit of small energy and large inter-brane
separation. Thus, we shall concentrate on the lowest order of the energy of
the radiated string and the large distance of the interacting branes.

To compare our results with Ref. [31], we apply the condition
$\mathcal{S}=\mathbf{1}$, which prominently imposes some restrictions on the
parameters of the setup. This constraint enables us to recast the two-
derivative term as a combination of the one-derivative terms. In the
calculations we receive a non-integral surface term at $\tau,t^{\prime}=0$.
Since it has no physical interpretation, it can be eliminated by assuming
$p_{\mu}p_{\nu}S_{1}^{\mu\nu}$, $p_{\mu}p_{\nu}S_{2}^{\mu\nu}<0$, [31, 32].
Thus, we obtain

$\langle\partial X^{\mu}\bar{\partial}X^{\nu}\rangle_{\rm
osc}|_{t\rightarrow\infty}=-\langle\partial X^{\mu}X^{\nu}\rangle_{\rm
osc}|_{t\rightarrow\infty}\left[\langle p\cdot\partial Xp\cdot X\rangle_{\rm
osc}|_{t\rightarrow\infty}+\dfrac{i\alpha^{\prime}}{2}(k_{1}^{2}-k_{2}^{2})\right].$
(4.1)

In addition, the integration by part gives the following equivalent relations

$\displaystyle\
\frac{e^{-4(\tau,t^{\prime})}}{1-e^{-4(\tau,t^{\prime})}}\left(1+\frac{1}{1-e^{-4(\tau,t^{\prime})}}\right)$
$\displaystyle\doteq$
$\displaystyle\frac{\xi-\alpha^{\prime}k_{(2,1)}^{2}}{20\alpha^{\prime}p_{\mu}p_{\nu}S_{(1,2)}^{\mu\nu}}.$
(4.2)

These relations reduce the dependency of the trace-terms to the proper times
$t^{\prime}$ and $\tau$. Finally, the integrals over the proper times, in the
eikonal approximation, is given by [33],

$\displaystyle\int_{0}^{\infty}{\rm d}t^{\prime}\int_{0}^{\infty}{\rm d}\tau\
e^{-\alpha^{\prime}k_{1}^{2}t^{\prime}}e^{-\alpha^{\prime}k_{2}^{2}\tau}e^{\xi(t^{\prime}+\tau)}\langle
e^{ip\cdot X_{\rm
osc}}\rangle|_{t\rightarrow\infty}^{{\mathcal{S}}=\mathbf{1}}\approx\Big{[}(\alpha^{\prime}k_{1}^{2}-\xi)(\alpha^{\prime}k_{2}^{2}-\xi)\Big{]}^{-1}.$
(4.3)

The polarization tensor of the Kalb-Ramond state has the properties
$\epsilon^{\rm KB}_{\mu\nu}=-\epsilon^{\rm KB}_{\nu\mu}$ and
$p^{\mu}\epsilon^{\rm KB}_{\mu\nu}=0$. Adding all these together yields the
amplitude of the Kalb-Ramond radiation

$\displaystyle\mathcal{A}^{\rm KB}_{\rm
R}|^{\mathcal{S}=\mathbf{1}}_{t\rightarrow\infty}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{p}T_{p}^{2}}{128\alpha^{\prime
2}(2\pi)^{7-p}}\epsilon_{ij}^{\rm
KB}\left(\prod_{\alpha=0}^{p}\delta(p^{\alpha})\right)\int_{-\infty}^{+\infty}\prod_{i=p+1}^{9}\frac{dk_{1}^{i}}{k_{1}^{2}k_{2}^{2}}e^{ik_{1}^{i}b^{i}}$
(4.4) $\displaystyle\times$
$\displaystyle\Bigg{\\{}i\mathbf{T}^{\prime-1}p_{k}p_{l}\left[\mathcal{T}^{ki}\mathcal{T}^{lj}+(-1)^{p+1}\mathcal{T}^{ki}\mathcal{T}^{\prime\
lj}-\mathcal{T}^{kl}\mathcal{T}^{ij}\right]$ $\displaystyle-$
$\displaystyle(2\mathbf{T})^{-1}p_{k}p_{l}\left[\mathcal{T}^{\prime\
kl}\mathcal{T}^{\prime\
ij}+(-1)^{p}\mathcal{T}^{ki}\mathcal{T}^{lj}-\mathcal{T}^{\prime\
ki}\mathcal{T}^{\prime\ lj}\right]$ $\displaystyle+$ $\displaystyle
4i\alpha^{\prime}p_{k}\Bigg{[}k_{1}^{i}\left(i\mathcal{T}^{\prime\
kj}-\frac{1}{2}\mathcal{T}^{kj}\right)-(-1)^{p}k_{1}^{j}\left(i\mathcal{T}^{\prime\
ki}-\frac{1}{2}\mathcal{T}^{ki}\right)\Bigg{]}\Bigg{\\}},$
$\displaystyle\mathcal{A}^{\rm KB}_{\rm
NS}|^{\mathcal{S}=\mathbf{1}}_{t\rightarrow\infty}$ $\displaystyle=$
$\displaystyle 0.$ (4.5)

We observe that, because of the anti-symmetric property of the polarization
tensor of the Kalb-Ramond state, the NS-NS sector does not possess any
contribution to the radiation amplitude.

Our results, which were acquired in the eikonal approximation, are comparable
with the results of Ref. [31]. In fact, the emission amplitude for any
massless state in the large distance limit generally includes one of the
following processes. The terms with the propagator factors $1/k_{1}^{2}$ and
$1/k_{2}^{2}$ elaborate that one of the branes emits a massless closed string,
then it is absorbed by the other brane, afterward it moves on the absorber
brane for a while, the absorber brane finally emits the final massless string
state. This kind of string radiation is similar to the bremsstrahlung process.
In the emission amplitude, the term with the factor $1/k_{1}^{2}k_{2}^{2}$
represents a double-pole mechanism, in which the massless string radiation
occurs from the exchanged closed string between the two D-branes. The latter
string obviously causes the interaction between the two D-branes.

We observed that the massless string radiation in the middle region of the
distant branes exclusively is for the Kalb-Ramond emission. This kind of
emission has not occurred in Ref. [31], and not even in the bosonic
configurations [33]. However, the absence of the terms with the propagator
factors $1/k^{2}_{1}$ and $1/k^{2}_{2}$ in the radiation amplitude (4.4)
manifestly demonstrates that there is no bremsstrahlung-like process.
Precisely, we do not have any massless string radiation from the surfaces of
the branes.

##  5 Conclusions

In the first part of the paper, we employed the boundary states formalism to
compute the radiation amplitude of a general massless closed string from the
interaction of two parallel D$p$-branes in the context of the superstring
theory. We assumed that the branes have been dressed with the Kalb-Ramond
field and different $U(1)$ gauge potentials.

In the second part of the paper, we specifically calculated the radiation
amplitude of the Kalb-Ramond state from the branes with large distance. For
this case, we imposed some constraints on the setup parameters. We observed
that the Kalb-Ramond radiation occurs only in a single physical process, i.e.
the radiation from the exchanged closed string between the interacting branes.
This radiation exclusively happens in the R-R sector, and the NS-NS sector
does not possess any contribution to it.

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# Computing aberration coefficients for plane-symmetric reflective systems: A
Lie algebraic approach

A. Barion  1,* M. J. H. Anthonissen  1 J. H. M. ten Thije Boonkkamp  1 and W.
L. IJzerman1,2  1Eindhoven University of Technology, PO Box 513, 5600 MB
Eindhoven, The Netherlands
2Signify, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands
<EMAIL_ADDRESS>
###### Abstract

We apply the Lie algebraic method to reflecting optical systems with plane-
symmetric freeform mirrors. Using analytical ray-tracing equations we
construct an optical map. The expansion of this map gives us the aberration
coefficients in terms of initial ray coordinates. The Lie algebraic method is
applied to treat aberrations up to arbitrary order. The presented method
provides a systematic and rigorous approach to the derivation, treatment and
composition of aberrations in plane-symmetric systems. We give the results for
second- and third-order aberrations and apply them to three single-mirror
examples.

††journal: opticajournal††articletype: Research Article

##  1 Introduction

Off-axis mirror systems provide additional degrees of freedom for the design
of more compact and accurate imaging systems compared to rotationally
symmetric ones. The study of their aberration behaviour is of great interest
to the optics community. The mathematical characterization of aberrations has
been investigated in [1, 2]. Possible approaches to derive explicit aberration
expansions are given in [3], where only confocal arrangements are considered,
or in [4] where the starting design is rotationally symmetric. Recently,
explicit expressions for plane-symmetric reflective optical systems have been
determined using a matrix formalism [5, 6, 7]. Here, the matrix method for
paraxial ray-tracing is extended to accommodate for higher degree polynomial
terms and aberrations are composed by manipulating the respective matrix
coefficients.

In this work we will describe the Lie algebraic method needed to obtain
analytical expressions for the aforementioned aberration terms of arbitrary
order. Starting from a chosen ray, which we will define as the optical axis
ray (OAR), we will follow its path through the system from object to image
plane. At the image plane the aberration terms are given as polynomials in
$(\bm{q},\bm{p})$, which are the phase-space variables of our optical system
[8, 9]. In this Hamiltonian formulation, the propagation and reflection maps
are symplectic, i.e., volume preserving in phase-space. Our goal is to
approximate these maps while preserving symplecticity. Applying these
approximating maps to the initial coordinates will deliver the desired
aberration expansion terms.

The Lie approach provides the tools to systematically determine the
approximating map for one single plane-symmetric mirror. The description of a
complete optical system is then reduced to a concatenation of maps. This
process is described and handled by the Lie theory. Compared to the matrix
formalism in [5, 6, 7], the mathematical framework of the Lie method reduces
the number of coefficients necessary to be stored. Additionally, the
phenomenon of low-order aberrations composing into higher order contributions
follows directly from the mathematical framework. This is also known as the
distinction between intrinsic and extrinsic aberrations [10], where low-order
aberrations of individual surfaces (intrinsic) combine into higher order
contributions to the complete system (extrinsic).

In Section 2 we will describe the explicit maps that govern ray propagation
and reflection in a plane-symmetric reflective optical system. A brief summary
of the essential Lie algebraic notions is given in Section 3, even though we
refer to [11, 9] for a more in depth description. Section 4 contains the steps
needed to construct the approximation maps and the calculations up to third-
order aberrations. Three examples to validate the presented method are given
in Section 5, where both existing theoretical and computational results are
reproduced.

##  2 Analytic Ray-Tracing

In this section we discuss the mappings needed to ray-trace light rays through
a reflective system composed of plane-symmetric, i.e., symmetric with respect
to the $yz$-plane, optical surfaces; see Figure 1. In order to follow a ray
path from object to image plane, we describe three transformations. First, the
incoming ray is propagated from the object plane to the reflecting surface and
the reflected ray from the surface to the image plane. Second, we describe the
reflection of the ray at a plane-symmetric mirror. Finally, rotation of the
coordinate system is shown, such that the $z$-axis remains aligned with the
optical axis ray (OAR) before and after reflection. This implies that the
considered $z$-axis will be broken into line segments.

Once these three mappings have been described, they are concatenated to
describe a single mirror, which we call the fundamental element, according to
the following five steps: $i)$ propagation from object plane to mirror; $ii)$
rotation of the optical axis and corresponding coordinate system by an angle
$\theta$, equal to the incidence angle of the OAR; $iii)$ reflection of the
rays; $iv)$ second rotation of coordinate system by the angle $\theta$ and
$v)$ propagation from mirror to image plane.

Position coordinates of an arbitrary ray before and after reflection are
projected along the ray onto the two planes passing through the point of
impact of the OAR and orthogonal to it; see Figure 1. The incoming (outgoing)
plane, which is orthogonal to the incoming (outgoing) OAR, will be called the
incoming (outgoing) standard screen and the incoming (outgoing) position and
direction coordinates will be evaluated with respect to it. The incoming
standard screen is the $xy$-plane and the outgoing one is the
$x^{\prime}y^{\prime}$-plane, where the $x$ and $x^{\prime}$-axis are the
same; see Figure 1.

In the remaining part of this section the three elementary maps are described
independently from each other. Eventually, we concatenate them to describe a
complete mirror element as previously described. Each ray is characterized by
its position $\bm{q}=(q_{x},q_{y})$ and its direction $\bm{p}=(p_{x},p_{y})$
at a standard screen. As such, we use the phase-space coordinates
$(\bm{q},\bm{p})$ as our ray coordinates, cf. [12]. Note that the coordinates
of the OAR are at the origin of phase-space both before and after reflection,
i.e., the OAR will have coordinates $\bm{q}=\bm{0}=\bm{q}^{\prime}$ and
$\bm{p}=\bm{0}=\bm{p}^{\prime}$. In the descriptions to follow phase-space
coordinates $(\bm{q},\bm{p})$ are mapped to primed coordinates
$(\bm{q}^{\prime},\bm{p}^{\prime})$ by the respective mappings.

Figure 1: Point $A$ of the incoming ray in $xy$-coordinates is mapped to point
$B$ of the outgoing ray in the rotated $x^{\prime}y^{\prime}$-coordinate
system. The axes $x$ and $x^{\prime}$ are perpendicular to the $yz$-plane (not
shown) and the incidence angle of the OAR is equal to $\theta$.

###  2.1 Propagation

We introduce the Hamiltonian $H(\bm{p})$ governing free propagation of light
in a medium of constant refractive index $n$ [13, 9, 12], i.e.,

$H(\bm{p})=-\sigma\sqrt{n^{2}-|\bm{p}|^{2}}=-\sigma p_{z},$ (2.1)

where $\bm{p}=(p_{x},p_{y})$ and $p_{z}$ are the direction momenta – direction
cosines times the refractive index $n$ – along the respective axes. The
variable $\sigma=\pm 1$, is positive for forward travelling rays and negative
for backwards propagating rays. Since we are only considering reflections, the
refractive index of our medium (air/vacuum) $n=1$. The distance measured along
the optical axis, which coincides with the $z$-axis, serves as evolution
parameter of the Hamiltonian system related to Eq. (2.1):

$\dot{\bm{q}}=\frac{\partial
H}{\partial\bm{p}}=-\frac{\bm{p}}{H},\qquad\dot{\bm{p}}=-\frac{\partial
H}{\partial\bm{q}}=\bm{0}.$ (2.2)

The solution to the Hamiltonian system Eq. (2.2) with initial conditions
$(\bm{q},\bm{p})$, after propagating a distance $d$ along the optical axis
ray, reads:

$\bm{q}^{\prime}=\bm{q}-d\frac{\bm{p}}{H(\bm{p})},\quad\bm{p}^{\prime}=\bm{p}.$
(2.3)

###  2.2 Reflection

Next, we consider the law of reflection in vector form regardless of the
coordinate system [14]

$\hat{\bm{k}}_{\mathrm{r}}=\hat{\bm{k}}_{\mathrm{i}}-2(\hat{\bm{k}}_{\mathrm{i}}\cdot\hat{\bm{n}})\hat{\bm{n}},$
(2.4)

where $\hat{\bm{k}}_{\mathrm{r}}$ is the unit direction vector of the
reflected ray, $\hat{\bm{k}}_{\mathrm{i}}$ the unit direction vector of the
incoming ray and $\hat{\bm{n}}$ the unit outer normal of the reflector at the
impact point. Here, $\hat{}$ (hat) indicates that the vector has length one
and with the term ‘outer’ we mean opposite to the incoming ray direction,
i.e., $\hat{\bm{k}}_{\mathrm{i}}\cdot\hat{\bm{n}}<0$. Let the reflector be
described by $z=\zeta(\bm{q})$, then the outer normal $\hat{\bm{n}}$ of the
surface at point $(\bm{q},\zeta(\bm{q}))$ reads

$\hat{\bm{n}}=\frac{(\nabla\zeta(\bm{q}),-1)}{\sqrt{1+|\nabla\zeta(\bm{q})|^{2}}}.$
(2.5)

The incoming and outgoing ray directions are
$\hat{\bm{k}}_{\mathrm{i}}=(\bm{p},p_{z})/n$ and
$\hat{\bm{k}}_{r}=(\bm{p}^{\prime},p_{z}^{\prime})/n$, respectively. The
vector $\hat{\bm{k}}_{\mathrm{r}}$ is calculated inserting Eq. (2.5) in Eq.
(2.4). This way we get for the reflected momenta
$(\bm{p}^{\prime},p_{z}^{\prime})$:

$\bm{p}^{\prime}=\bm{p}-2\frac{\nabla\zeta(\bar{\bm{q}})}{1+|\nabla\zeta(\bar{\bm{q}})|^{2}}(\bm{p}\cdot\nabla\zeta(\bar{\bm{q}})-p_{z}),$
(2.6a)
$p_{z}^{\prime}=p_{z}+\frac{2}{1+|\nabla\zeta(\bar{\bm{q}})|^{2}}(\bm{p}\cdot\nabla\zeta(\bar{\bm{q}})-p_{z}),$
(2.6b)

where $(\bar{\bm{q}},\zeta(\bar{\bm{q}}))$ is the intersection point of the
incoming ray and the reflector. The intersection point $\bar{\bm{q}}$ is
related to the screen coordinate before reflection $\bm{q}$ and the one after
reflection $\bm{q}^{\prime}$ by [9, 13, 12]

$\bar{\bm{q}}=\bm{q}+\zeta(\bar{\bm{q}})\frac{\bm{p}}{p_{z}},\quad\bm{q}^{\prime}=\bar{\bm{q}}-\zeta(\bar{\bm{q}})\frac{\bm{p}^{\prime}}{p_{z}^{\prime}}.$
(2.7)

Eq. (2.7) gives an implicit relation for $\bar{\bm{q}}$ which needs to be
solved iteratively; see [12, 15, 13]. The Eqs. (2.7) are again solution to the
Hamiltonian system Eq. (2.2), but now propagating a distance
$d=\zeta(\bar{\bm{q}})$.

###  2.3 Rotation of the Standard Screen

After propagation and reflection, we discuss the necessary steps to rotate our
coordinates according to the OAR. We describe an arbitrary rotation by an
angle $\theta$ that rotates our standard screen, see Figure 2. For a single
reflector two rotations of angle $\theta$ are used, where eventually $\theta$
is the incidence angle of the OAR. The first rotation brings the $z$-axis of
the incoming coordinate system from being aligned with the OAR to being
aligned with the surface normal at the point of intersection of the OAR. The
surface equation $z=\zeta(\bm{q})$ is defined in this coordinate system
aligned with its normal and therefore has zero gradient at the origin, i.e.,
$\nabla\zeta(\bm{0})=\bm{0}$, which is the point of impact of the OAR. We then
apply the reflection mapping and subsequently rotate to the outgoing
coordinate system aligned with the reflected OAR, see Figure 3.

Let us define positive rotations when the $y$-axis is rotated towards the
$z$-axis (clock-wise). In the starting coordinate system the $z$-axis is
aligned with the incoming OAR and as such we can call it the incoming
coordinate system. We consider surfaces with plane-symmetry with respect to
the $yz$-plane and as such the rotations are around the $x$-axis.

The rotation mapping of the screen around the $x$-axis for the phase-space
coordinates can be found as a Lie transformation [9]. Here we present an
equivalent derivation. The momentum coordinates $p_{x},p_{y},p_{z}$ are
rotated into the coordinates $p_{x}^{\prime},p_{y}^{\prime},p_{z}^{\prime}$
according to the well-known rotation matrix

$\begin{pmatrix}p_{x}^{\prime}\\\ p_{y}^{\prime}\\\
p_{z}^{\prime}\end{pmatrix}=\begin{pmatrix}1&0&0\\\ 0&\cos\theta&\sin\theta\\\
0&-\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}p_{x}\\\ p_{y}\\\
p_{z}\end{pmatrix}.$ (2.8)

The expressions for the position coordinates are more complex. In fact, recall
that we map the intersection points of the light rays with the (rotated)
standard screens; see Figure 2.

Figure 2: Upon rotation of axes we map the position coordinate $q_{y}$ to the
position coordinate $q_{y}^{\prime}$ relating to the same ray (red).

As such, let us first fix the parametrization of the ray and the normal
equation of the tilted surface. The path of the ray can be parametrized by

$\binom{\bm{q}}{0}+\lambda\binom{\bm{p}}{p_{z}},\quad\lambda\in\mathbb{R}.$
(2.9)

After the first rotation, the equation of the rotated standard screen, with
normal $(0,\sin\theta,-\cos\theta)$ and passing through $(0,0,0)$, reads

$y\sin\theta-z\cos\theta=0.$ (2.10)

By substituting the parametrization (2.9) into the rotated standard screen
equation (2.10) we can solve for $\lambda$ to get the point of intersection.
We get

$\lambda=\frac{q_{y}\sin\theta}{p_{z}\cos\theta-p_{y}\sin\theta}.$ (2.11)

Substituting the value in Eq. (2.11) in the parametrization (2.9) gives us the
coordinates of the point of intersection of the considered ray and the tilted
screen. The last step is to derive the position coordinates with respect to
the rotated coordinate system, which corresponds to dividing the
$y$-coordinate by $\cos\theta$. The map for positive rotation around the
$x$-axis of the position coordinates $\bm{q}$ by an angle $\theta$ reads:

$q_{x}^{\prime}=\frac{q_{x}\,p_{z}\cos\theta-(q_{x}\,p_{y}-q_{y}\,p_{x})\sin\theta}{p_{z}\cos\theta-
p_{y}\sin\theta},\\\ $ (2.12a)
$q_{y}^{\prime}=\frac{q_{y}\,p_{z}}{p_{z}\cos\theta-p_{y}\sin\theta}.$ (2.12b)

In Eq. (2.12) the condition $p_{z}\cos\theta-p_{y}\sin\theta=0$ implies that
the considered ray is parallel to the rotated plane and as such will not
intersect with the plane.

###  2.4 The Fundamental Map

With the transformations described in Eqs. (2.6)-(2.8) and (2.12) we can
rotate the coordinate system for the $z$-axis to be aligned with the surface
normal, reflect the incoming rays and rotate the system again to align the
$z$-axis with the outgoing OAR. If propagation before and after the surface
are added to this map, we will call it the fundamental map. This composition
of transformations can be expanded up to the desired order in terms of the
phase-space coordinates $(\bm{q},\bm{p})$.

After reflection, $\sigma$ in the Hamiltonian described in Eq. (2.1) changes
sign. An intuitive way to understand this is to recall Eq. (2.1) with
$\sigma=1$ where $H=-p_{z}$. By the condition
$\hat{\bm{k}}_{\mathrm{i}}\cdot\hat{\bm{n}}<0$ that we imposed at reflection,
the reflected OAR travels in the same $z$-direction as the surface normal,
which is opposite to the one of the incoming OAR. This would lead to negative
propagation distances. Since we prefer to consider forward moving rays, we opt
for dealing with a left-handed coordinate system and align the
$z^{\prime}$-axis in Figure 3 with the direction of the reflected OAR after
the second rotation. It can be verified that this change does not influence
the form of our rotation map and the reflection map remains also unchanged.
The only important caveats are that the reflective surface must always be
described in the coordinate system of the incoming OAR and that angles are
positive when the $y$-axis rotates towards the $z$-axis.

Figure 3: Definition of incoming $xyz$ and outgoing
$x^{\prime}y^{\prime}z^{\prime}$-coordinate systems. The auxiliary system
denoted by the index $s$ is where the surface equation is defined.

With the mappings described in Eqs. (2.6)-(2.8) and (2.12) we can concatenate
them into a reflection plus rotation mapping. We define this composition of
transformations by $\mathcal{S}(\theta)$. Let $\mathcal{R}(\theta)$ denote the
rotation mapping by an angle of $\theta$ and $\mathcal{T}$ the reflection
mapping. Then, we can concisely describe the map $\mathcal{S}(\theta)$ as

$\mathcal{S}(\theta)=\mathcal{R}(\theta)\,\mathcal{T}\,\mathcal{R}(\theta).$
(2.13)

In Figure 1, we have that $\mathcal{S}(\theta)$ maps $A$ to $B$. This
definition of $\mathcal{S}(\theta)$ is necessary to apply the Lie algebraic
method. Note that the surface equation is given with respect to the coordinate
system denoted by the index $s$ in Figure 3.

To conclude, concatenating $\mathcal{S}(\theta)$ with propagation in object
and image-space, $\mathcal{P}_{\mathrm{ob}}$ and $\mathcal{P}_{\mathrm{im}}$
respectively, constitutes the fundamental map $\mathcal{M}$ necessary for our
description of the optical system

$\mathcal{M}=\mathcal{P}_{\mathrm{im}}\mathcal{S}(\theta)\mathcal{P}_{\mathrm{ob}}.$
(2.14)

##  3 Lie Algebraic Tools

With the help of the Lie algebraic method it is possible to construct
operators that reproduce the actions of propagation, reflection and rotation.
These operators enable us to derive closed form expressions for the aberration
components of an arbitrary optical system. A more detailed description of the
Lie algebraic tools used in this work can be found in [12, 9, 13]. Here, we
briefly introduce the main concepts.

The space of functions on phase-space becomes a Lie algebra when endowed with
the Poisson bracket $[\cdot,\cdot]$. The Poisson bracket of two functions
$f(\bm{q},\bm{p}),g(\bm{q},\bm{p})$ is defined as

$[f,g]=\frac{\partial f}{\partial\bm{q}}\bm{\cdot}\frac{\partial
g}{\partial\bm{p}}-\frac{\partial f}{\partial\bm{p}}\bm{\cdot}\frac{\partial
g}{\partial\bm{q}}.$ (3.1)

Accordingly, we can associate with each $f$ a Lie operator $[f,\cdot\,]$ that
acts on a second function $g$ by taking the Poisson bracket of the two. For
example, $[q_{1},\cdot\,]=\partial\cdot/\partial p_{1}$ and for vectors we
have $[\bm{q},\cdot\,]=\partial\cdot/\partial\bm{p}$.

Using the Poisson bracket, we can associate to each function $f$ on phase-
space a mapping $\exp([f,\cdot\,])$, called a Lie transformation, defined as

$\exp([f,\cdot\,])=\sum_{k=0}^{\infty}\frac{[f,\cdot\,]^{k}}{k!},$ (3.2)

where $[f,\cdot\,]^{0}=I$ and $[f,\cdot\,]^{k}=[f,[f,\cdot\,]^{k-1}]$ for
$k>1$. Suppose $f$ is only dependent on $\bm{q}$, i.e. $f=f(\bm{q})$, then

$\exp([f(\bm{q}),\cdot\,])\bm{q}=\bm{q}\quad\text{and}\quad\exp([f(\bm{q}),\cdot\,])\bm{p}=\bm{p}+\frac{\partial
f}{\partial\bm{q}}.$ (3.3)

In Eq. (3.3) the infinite series is truncated after the first two terms as any
subsequent one is equal to zero. Note that Lie transformation are applied
component-wise to vectors.

A map $(\bm{q},\bm{p})\mapsto(\bm{q}^{\prime}(\bm{q},\bm{p})$,
$\bm{p}^{\prime}(\bm{q},\bm{p}))$ is said to be a symplectic transformation,
if it satisfies [9, 11]:

$\displaystyle[q^{\prime}_{i},q^{\prime}_{j}]$
$\displaystyle=[q_{i},q_{j}]=0,$
$\displaystyle[p^{\prime}_{i},p^{\prime}_{j}]$
$\displaystyle=[p_{i},p_{j}]=0,$ (3.4)
$\displaystyle[q^{\prime}_{i},p^{\prime}_{j}]$
$\displaystyle=[q_{i},p_{j}]=\delta_{ij},$

where $\delta_{ij}$ is the Kronecker delta. Symplectic transformations
preserve volumes in phase-space. In fact, light propagation, reflection and
rotation are all symplectic maps.

It can be proven that a mapping defined as in Eq. (3.2) is symplectic [11].
Conversely, symplectic mappings $\mathcal{M}$ that map the origin to itself,
i.e., $\mathcal{M}(\bm{0})=\bm{0}$, can be represented as an infinite
concatenation of Lie transformations of the form

$\mathcal{M}=\exp([g_{2},\cdot\,])\exp([g_{3},\cdot\,])\cdots,$ (3.5)

where the generators $g_{2},g_{3},$ etc. are homogeneous polynomials in the
variables $(\bm{q},\bm{p})$ of degree $2,3,$ etc. [11]. Here, we omit the
concatenation symbol $\circ$, as it is clear from the context that we are
concatenating operators. Recall that a homogeneous polynomial $g$ of degree
$m$, as in Eq. (3.5), has the following property

$g(\lambda\bm{q},\lambda\bm{p})=\lambda^{m}g(\bm{q},\bm{p})\quad\forall\lambda\in\mathbb{R}.$
(3.6)

The maps for ray propagation and reflection plus rotation are symplectic and
map the origin onto itself [9, 13, 12]. It is therefore possible to represent
them as an infinite, or approximate them by a truncated, concatenation of Lie
transformations according to the result in Eq. (3.5) and then rearrange the
Lie transformations using additional Lie tools given in Appendix A, cf. Eq.
(A.1) and Eq. (A.2).

Our aim is to approximate the fundamental map $\mathcal{M}$ in Eq. (2.14) of a
reflector by means of a truncated concatenation of Lie transformations in
ascending order, similarly to the structure of Eq. (3.5). This enables us to
clearly distinguish which parts of the map influence which order of
aberrations. In fact, generators of order $k$ are directly related to the
transverse ray aberrations of order $k-1$ [12]. Concatenating multiple
fundamental maps representing the different mirrors in our system and
disregarding terms that lead to higher order aberrations leads to a map
describing the complete optical system – up to the desired order of accuracy
in terms of initial phase-space coordinates.

##  4 The Fundamental Element

Free propagation, reflection and rotation are symplectic maps and their
combined actions map the origin of phase-space, i.e., the OAR, to itself. As
such, it is possible to represent the combined actions of reflection and
rotation $\mathcal{S}(\theta)$, see Eq. (2.13), in the form of Eq. (3.5). The
polynomials necessary for this representation in terms of Lie transformations
are called the generators of the map. It is important to consider the complete
reflection with rotation map $\mathcal{S}(\theta)$ because this ensures that
the origin of phase-space, i.e., our OAR, is mapped onto itself. Hence, we
have that $(\mathcal{S}(\theta))(\bm{0})=\bm{0}$ and we can apply the results
in Eq. (3.5). Rotation alone does not map the origin of phase-space onto
itself.

We subsequently concatenate $\mathcal{S}(\theta)$ with the maps of object and
image-space propagation to derive the description of the fundamental element
of the optical system. The fundamental element represents the physical
counterpart of the fundamental map described at the end of Section 2. This
fundamental element is the building block of any arbitrary reflecting optical
system with plane-symmetry with respect to the $yz$-plane. We restrict our
analysis to aberrations of order three and therefore only polynomials up to
degree four in Eq. (3.5) are of relevance; see [12, 13, 9].

The generators of free propagation for light rays in a medium of refractive
index $n=1$ are, up to degree four [12],

$h_{2}(\bm{p})=\frac{1}{2}|\bm{p}|^{2},\quad
h_{4}(\bm{p})=\frac{1}{8}|\bm{p}|^{4}.$ (4.1)

This means, that if we want to propagate our physical system with initial
condition $(\bm{q},\bm{p})$ over a distance $d$ along the optical axis, then
the expression

$\binom{\bm{q}^{\prime}}{\bm{p}^{\prime}}=\exp(-d[h_{2},\cdot\,])\exp(-d[h_{4},\cdot\,])\binom{\bm{q}}{\bm{p}},$
(4.2)

is equal, up to third-order terms, to the solution given in Eq. (2.3) [12, 13,
9]. The result of Eq. (4.2) is therefore sufficiently accurate to investigate
third-order aberrations. Note that the polynomials in Eq. (4.1) are simply the
first two terms in the Taylor expansion of the Hamiltonian defined in Eq.
(2.1).

The mirror equation is given in the coordinate system with its $z$-axis
aligned with the surface’s normal and is of the form

$z=\zeta(\bm{q})=\sum_{\begin{subarray}{c}2\leq m+n\leq 4\\\ m\text{
even}\end{subarray}}c_{mn}q_{x}^{m}q_{y}^{n}.$ (4.3)

We consider surface terms up to fourth order, since higher order terms do not
influence third-order aberrations.

The reflection and rotation mapping $\mathcal{S}(\theta)$ maps $\bm{q},\bm{p}$
to $\bm{q}^{\prime},\bm{p}^{\prime}$. First, the rotation by the angle
$\theta$ is applied to the incoming ray coordinates, cf. Eqs. (2.8),(2.12).
Secondly, reflection acts on these already rotated coordinates, cf. Eqs.
(2.6),(2.7). Lastly, a second rotation by $\theta$ maps these coordinates into
the final reflected coordinate system. All these transformations – and their
concatenation – can be expanded in terms of $(\bm{q},\bm{p})$ with the aid of
computer algebra software, e.g., Mathematica. The first order expansion of
$\mathcal{S}(\theta)$ reads:

$\displaystyle q^{\prime}_{x}$ $\displaystyle=q_{x},$ (4.4) $\displaystyle
q^{\prime}_{y}$ $\displaystyle=q_{y},$ $\displaystyle p^{\prime}_{x}$
$\displaystyle=p_{x}+4\,c_{20}\cos(\theta)\,q_{x},$ $\displaystyle
p^{\prime}_{y}$ $\displaystyle=p_{y}+4\,c_{02}\sec(\theta)\,q_{y}.$

Here, the coefficients $c_{20},c_{02}$, cf. Eq. (4.3), can be related to the
radii of curvature of the mirror surface. The polynomial $g_{2}$ associated
with the Lie transformation that generates the linear map in Eq. (4.4) is only
dependent on $\bm{q}$:

$g_{2}(\bm{q})=2\,c_{20}\cos(\theta)\,q_{x}^{2}+2\,c_{02}\sec(\theta)\,q_{y}^{2}.$
(4.5)

The Lie transformation generated by Eq. (4.5) reads

$\binom{\bm{q}^{\prime}}{\bm{p}^{\prime}}=\exp([g_{2}(\bm{q}),\cdot\,])\binom{\bm{q}}{\bm{p}}=\binom{\bm{q}}{\bm{p}+\dfrac{\partial
g_{2}(\bm{q})}{\partial\bm{q}}}.$ (4.6)

One can verify that the expression in Eq. (4.6) is the same as the one in Eq.
(4.4). Note that, if $g_{2}$ would also depend on $\bm{p}$, it would generate
contributions to the $\bm{q}$-coordinates, which is undesired; cf. Eq. (4.4).

To initiate a more systematic approach, we define the generators $g_{m}$ in a
more general way:

$g_{m}(\bm{q},\bm{p})=\sum_{i+j+k+l=m}a_{ijkl}\,q_{x}^{i}\,q_{y}^{j}\,p_{x}^{k}\,p_{y}^{l},\quad
i,j,k,l\in\mathbb{N},\quad i+k\text{ even}$ (4.7)

where the condition $i+k$ even stems from the symmetry of the optical system
itself. In this notation the functions $g_{2},g_{3},g_{4}$ are defined by
their coefficients. It is our goal to determine these coefficients such that

$\mathcal{S}(\theta)\overset{(3)}{=}\exp([g_{2},\cdot\,])\exp([g_{3},\cdot\,])\exp([g_{4},\cdot\,]).$
(4.8)

The notation $\overset{(3)}{=}$ symbolizes that the truncated concatenation of
Lie transformations on the right-hand side (RHS) of Eq. (4.8) produces the
same expressions as the map $\mathcal{S}(\theta)$ up to third-order terms in
phase-space coordinates.

To derive these coefficients, one has to expand the mapping
$\mathcal{S}(\theta)$ up to the order of interest, i.e., 3 in our case.
Subsequently, we consider the general form of the generators as given in Eq.
(4.7) and compute the action of the concatenation of Lie transformations in
Eq. (4.8) on the phase-space variables. Since the coefficients of the
generator $g_{k}$ are fully determined by their contributions to the
aberrations of order $k-1$, we can determine the coefficients of the
generators in increasing order; see the method described in [12]. The non-zero
coefficients of the generators of the reflection plus rotation mapping
$\mathcal{S}(\theta)$ are listed in Table 1 in the form described by Eq.
(4.7).

Coefficients |  Values
---|---
$a_{0200}$ |  $2\,c_{02}\sec(\theta)$
$a_{2000}$ |  $2\,c_{20}\cos(\theta)$
$a_{0201}$ |  $2\,c_{02}\sec(\theta)\tan(\theta)$
$a_{0300}$ |  $2\sec(\theta)^{2}(c_{03}-2\,c_{02}^{2}\tan(\theta))$
$a_{1110}$ |  $4\,c_{20}\sin(\theta)$
$a_{2001}$ |  $-2\,c_{20}\sin(\theta)$
$a_{2100}$ |  $2(c_{21}-4\cos(\theta)\,c_{20}^{2}\sin(\theta)+2\,c_{02}\,c_{20}\tan(\theta))$
$a_{0202}$ |  $-\frac{1}{2}(-1+3\cos(2\theta))\,c_{02}\sec(\theta)^{3}$
$a_{0220}$ |  $-c_{02}\sec(\theta)+2\,c_{20}\sin(\theta)\tan(\theta)$
$a_{0301}$ |  $2\sec(\theta)^{4}(-\,c_{02}^{2}+3\cos(2\theta)\,c_{02}^{2}+\,c_{03}\sin(2\theta))$
$a_{0400}$ |  $-\sec(\theta)^{5}(\,c_{02}^{3}+\cos(2\theta)(7\,c_{02}^{3}-c_{04})-c_{04}+4\,c_{02}\,c_{03}\sin(2\theta))$
$a_{1210}$ |  $4\,c_{02}\,c_{20}-2\,c_{20}^{2}+2\cos(3\theta)\,c_{20}^{2}\sec(\theta)+4\,c_{21}\tan(\theta)$
$a_{2002}$ |  $-\cos(\theta)\,c_{20}$
$a_{2020}$ |  $-\cos(\theta)\,c_{20}$
$a_{2101}$ |  $4\,c_{02}\,c_{20}$
$a_{2200}$ |  $2(-4\cos(\theta)\,c_{02}\,c_{20}^{2}+8\,c_{02}^{2}\,c_{20}\sec(\theta)^{3}-4\,c_{20}\,c_{21}\sin(\theta)+\sec(\theta)(-12\,c_{02}^{2}\,c_{20}+c_{22}-6\,c_{02}^{2}\,c_{20}\tan(\theta)^{2}))$
$a_{3010}$ |  $4\cos(\theta)^{2}\,c_{20}^{2}$
$a_{4000}$ |  $-8\cos(\theta)^{3}\,c_{20}^{3}+2\cos(\theta)\,c_{40}-2\,c_{02}\,c_{20}^{2}\sin(\theta)\tan(\theta)$
Table 1: Coefficients of $g_{2},g_{3},g_{4}$.

We proceed to combine reflection and rotation with propagation before and
after the surface. The mapping $\mathcal{M}$ will describe the action of a
fundamental element on the rays from object plane coordinates
$(\bm{q},\bm{p})$ to image plane coordinates
$(\bm{q}^{\prime},\bm{p}^{\prime})$. The map $\mathcal{M}$ is, up to fourth
degree generators, composed as follows:

$\mathcal{M}\overset{(3)}{=}\underbrace{\exp\left(-s_{\mathrm{ob}}[h_{2},\cdot\,]\right)\exp\left(-s_{\mathrm{ob}}[h_{4},\cdot\,]\right)}_{\overset{(3)}{=}\text{propagation
from object
plane}}\underbrace{\exp([g_{2},\cdot\,])\exp([g_{3},\cdot\,])\exp([g_{4},\cdot\,])}_{\overset{(3)}{=}\mathcal{S}(\theta)}\\\
\underbrace{\exp\left(-s_{\mathrm{im}}[h_{2},\cdot\,]\right)\exp\left(-s_{\mathrm{im}}[h_{4},\cdot\,]\right)}_{\overset{(3)}{=}\text{propagation
to image plane}}.$ (4.9)

Here, $s_{\mathrm{ob}},s_{\mathrm{im}}$ are the object and image distances
measured along the OAR in the sagittal plane. Although it might appear
counter-intuitive, due to their unique properties, the order of Lie
transformations is left-to-right like the order of transformations undergone
by the ray [9]. The object and image distances for the sagittal and tangential
planes satisfy the Coddington equations [16]:

$\displaystyle\text{sagittal plane}{:}\quad$
$\displaystyle\frac{1}{s_{\mathrm{ob}}}+\frac{1}{s_{\mathrm{im}}}=-4\,c_{20}\cos(\theta),$
(4.10a) $\displaystyle\text{tangential plane}{:}\quad$
$\displaystyle\frac{1}{t_{\mathrm{ob}}}+\frac{1}{t_{\mathrm{im}}}=-4\,c_{02}\sec(\theta).$
(4.10b)

We want to reorder and combine the Lie transformations of Eq. (4.9) into three
Lie transformations generated by the functions $\tau_{2},\tau_{3},\tau_{4}$
such that

$\mathcal{M}\overset{(3)}{=}\exp([\tau_{2},\cdot\,])\exp([\tau_{3},\cdot\,])\exp([\tau_{4},\cdot\,]).$
(4.11)

This allows us to separate the linear part of the mapping, generated by
$\tau_{2}$, from the higher order parts generated by $\tau_{3},\tau_{4}$ that
induce aberrations. Again, equality up to third-order expansions is sufficient
for our current work since we are investigating aberrations up to this same
order. The functions $\tau_{2},\tau_{3},\tau_{4}$ describe the action of a
fundamental element up to the expansion order $3$. To derive the functions
$\tau_{2},\tau_{3},\tau_{4}$ it is necessary to manipulate the mapping in Eq.
(4.9) such that the generators are combined and reordered in ascending order.
The procedure has been shown in [12] and a short example can be found in
Appendix A. The main tools necessary for these calculations are the Baker-
Campbell-Hausdorff (BCH) formula (A.1) and the identity given in Eq. (A.2).

The Lie transformation generated by the second degree polynomial $\tau_{2}$ is
more conveniently represented in its matrix form $M_{G}$ as it is the matrix
multiplication of its three components, i.e., object-space propagation,
reflection with rotation and image-space propagation. We call this the
Gaussian part of the mapping $\mathcal{M}_{G}=\exp([\tau_{2},\cdot\,])$ and
the associated $M_{G}$ reads

$M_{G}=\begin{pmatrix}1&0&s_{\mathrm{im}}&0\\\ 0&1&0&s_{\mathrm{im}}\\\
0&0&1&0\\\ 0&0&0&1\end{pmatrix}\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\
4\,c_{20}\cos(\theta)&0&1&0\\\
0&4\,c_{02}\sec(\theta)&0&1\end{pmatrix}\begin{pmatrix}1&0&s_{\mathrm{ob}}&0\\\
0&1&0&s_{\mathrm{ob}}\\\ 0&0&1&0\\\ 0&0&0&1\end{pmatrix}.$ (4.12)

The coefficients of the polynomial $\tau_{3}$ are given in Table 2 analogously
to Eq. (4.7) with coefficients denoted by $b_{ijkl}$.

Coefficients | Values
---|---
$b_{0003}$ | $s_{\mathrm{im}}^{2}\left(a_{0201}-s_{\mathrm{im}}a_{0300}\right)$
$b_{0021}$ | $s_{\mathrm{im}}^{2}\left(a_{1110}+a_{2001}-s_{\mathrm{im}}a_{2100}\right)$
$b_{0102}$ | $s_{\mathrm{im}}\left(3s_{\mathrm{im}}a_{0300}-2a_{0201}\right)$
$b_{0120}$ | $s_{\mathrm{im}}\left(s_{\mathrm{im}}a_{2100}-a_{1110}\right)$
$b_{0201}$ | $a_{0201}-3s_{\mathrm{im}}a_{0300}$
$b_{0300}$ | $a_{0300}$
$b_{1011}$ | $s_{\mathrm{im}}\left(2s_{\mathrm{im}}a_{2100}-a_{1110}+2a_{2001}\right)$
$b_{1110}$ | $a_{1110}-2s_{\mathrm{im}}a_{2100}$
$b_{2001}$ | $a_{2001}-s_{\mathrm{im}}a_{2100}$
$b_{2100}$ | $a_{2100}$
Table 2: Coefficients of $\tau_{3}$.

The expressions for the coefficients of $\tau_{4}$ are rather lengthy and not
useful for the current discussion, but can be found in Appendix B for
completeness. We thus have a mapping that describes the fundamental element up
to third-order.

###  4.1 From Optical Element to Optical System

To treat optical systems it suffices to concatenate multiple fundamental
elements, keeping in mind the sign conventions described at the end of Section
2.4. Each intermediate image plane corresponds to the intermediate object
plane of the subsequent mirror. Thus, if one fundamental element is described
by Eq. (4.11), then multiple elements are a concatenation of Lie
transformations of this form. For example, suppose a two-mirror system where
one mirror is described by the generators $\tau_{k}$ and the other mirror by
the generators $\sigma_{k}$. Then, the map $\mathcal{M}$ of the complete
system, up to third-order contributions, reads,

$\mathcal{M}\overset{(3)}{=}\exp([\tau_{2},\cdot\,])\exp([\tau_{3},\cdot\,])\exp([\tau_{4},\cdot\,])\exp([\sigma_{2},\cdot\,])\exp([\sigma_{3},\cdot\,])\exp([\sigma_{4},\cdot\,]).$
(4.13)

The coefficients of $\tau_{k},\sigma_{k}$ are completely described by the
geometry of the system according to the expressions $b_{ijkl}$.

Previously, we have stressed the importance of having the Lie transformations
in ascending order. This allows us to separate the contributions to the
different (ascending) orders of aberrations. The necessary computations to
reorder Eq. (4.13) rely on the procedure for reordering shown in [12] and make
use of the BCH formula (A.1) and the results of Eq. (A.2). During these steps,
the composition of low-order aberrations into high-order ones follows directly
from the application of the BCH formula.

In more complex optical systems the intermediate image planes for the sagittal
and tangential rays need not to be located at the same point along the OAR. As
such, the choice of the propagation distances for each fundamental element is
seemingly unclear. However, whatever the chosen propagation distance is equal
to, the sum of the intermediate image distance of the surface $j$ and the
object distance of surface $j+1$ needs always be equal to the total distance
between the two surfaces. Since the propagation mappings commute, see [12, 13,
9], it does not matter what distance is chosen for the image propagation of
surface $j$ or the object distance for surface $j+1$ as long as their sum
remains equal to the distance between the two surfaces.

##  5 Applications

We verify the presented methodology using three examples. We recover the
surface expansion coefficients of a spherical ellipsoid for a point-to-point
imager and the surface expansion coefficients for a focusing mirror as
recently presented in [5]. Lastly, we use our proposed method to ray-trace a
beam of rays reflected by a biconic mirror and compare with the spot diagram
generated using OpticStudio.

The first example will be the problem of perfect point-to-point imaging; see
Figure 4. Suppose we have an object point on the OAR which is then reflected
off a surface onto an image point. A spherical ellipsoid with these two points
at its foci will result in perfect imaging [17], i.e., no aberrations will be
present. Therefore, if we choose arbitrarily an object and an image point and
impose zero aberrations up to third-order for all rays with initial position
$\bm{q}^{\mathrm{ob}}=\bm{0}$, then the solution for the surface coefficients
should be the surface expansion terms up to fourth order of the corresponding
spherical ellipsoid.

Figure 4: A spherical ellipsoid as perfect imager between its foci. The OAR is
in red and the red dashed lines are other rays originating from the object.
The object and image planes are represented. The black dashed line is the
major axis of the ellipsoid.

We fix the object distance, image distance and the surface coefficients
$c_{20},c_{02}$ to have the desired paraxial properties. Subsequently, the
corresponding map given in Eq. (4.11) is applied to the initial coordinates
$(\bm{0},\bm{p}^{\mathrm{ob}})$. The expression for the final position
coordinates at the image plane $\bm{q}^{\mathrm{im}}$ is of the form:

$\bm{q}^{\mathrm{im}}=\bm{q}^{\mathrm{im}}(\bm{0},\bm{p}^{\mathrm{ob}}).$
(5.1)

Eq. (5.1) is a polynomial dependent only on the initial direction
$\bm{p}^{\mathrm{ob}}$ where each monomial coefficient will depend on the
chosen parameters and the – yet undetermined – higher order coefficients
$c_{mn}$ of the reflecting surface; see Eq. (4.3). The requirement of zero
aberration, i.e., $\bm{q}^{\mathrm{im}}=\bm{0}$, simply translates in setting
all monomial coefficients in Eq. (5.1) equal to zero. The resulting system of
equations will determine the value of the surface expansion coefficients. For
example, if we choose a spherical ellipsoid with major axis $a=20$ and minor
axis $b=10$, then with the corresponding initial parameters for the system
$s_{\mathrm{ob}}=s_{\mathrm{im}}=20$ and $c_{20}=-1/20,c_{02}=-1/80$, we get
the following system of equations for the unknown coefficients $c_{mn}$ with
$m=0,2,4$ and $2\leq m+n\leq 4$:

$\begin{dcases}80\,p_{x}^{3}(8000c_{40}+1)=0,\\\
-80\,p_{x}p_{y}^{2}\left(2400\sqrt{3}c_{03}+200\sqrt{3}c_{21}-16000c_{22}-1\right)=0,\\\
32000\,p_{x}p_{y}c_{21}=0,\\\
80\,p_{x}^{2}p_{y}\left(2400\sqrt{3}c_{03}+200\sqrt{3}c_{21}+16000c_{22}+1\right)=0,\\\
16000\,p_{x}^{2}c_{21}=0,\\\ 80\,p_{y}^{3}(128000c_{04}+1)=0,\\\
192000\,p_{y}^{2}c_{03}=0.\end{dcases}$ (5.2)

The solution to Eqs. (5.2), computed with exact arithmetic, gives the surface
expansion coefficients shown in Table 3.

$c_{mn}$ | Values
---|---
$c_{21}$ | 0
$c_{03}$ | 0
$c_{40}$ | $-1/8000$
$c_{22}$ | $-1/16000$
$c_{04}$ | $-1/128000$
Table 3: Surface expansion coefficients for the spherical ellipsoid defined
according to Eq. (4.3).

The coefficients in Table 3 are the same we would get by directly expanding
the ellipsoid’s equation

$z=\zeta(\bm{q})=\frac{b}{a}\sqrt{a^{2}-q_{y}^{2}-\frac{a^{2}}{b^{2}}q_{x}^{2}}-b,$
(5.3)

in terms of $q_{x},q_{y}$ around the origin. In fact, the surface equation
(5.3) represents the ellipsoid at the point of impact of the OAR with respect
to the coordinate system aligned with its normal at that point; see Figure 4.

The next example reproduces some of the results given in [5]. Here, the
authors calculate the surface expansion coefficients for a single mirror where
again zero third-order aberrations are imposed with the additional condition
that the initial momenta are equal to zero, i.e.,
$\bm{p}^{\mathrm{ob}}=\bm{0}$; see Figure 5.

Figure 5: A focusing reflector for an object at infinity as used in [5].

The initial surface parameters are the effective radius of curvature $R=-200$,
cf. [5], for both the sagittal and tangential planes and the incidence angle
of $\theta=-0.2$. The expression for the final position coordinates
$\bm{q}^{\mathrm{im}}$ is now a polynomial in $\bm{q}^{\mathrm{ob}}$, i.e.,

$\bm{q}^{\mathrm{im}}=\bm{q}^{\mathrm{im}}(\bm{q}^{\mathrm{ob}},\bm{0}).$
(5.4)

Setting all monomial coefficients of Eq. (5.4) to zero, results in the
following system of equations for the $c_{mn}$ with $m=0,2,4$ and $2\leq
m+n\leq 4$:

$\begin{dcases}\frac{q_{x}^{3}\left(-8R^{3}\cos(\theta)c_{40}+\sec^{2}(\theta)-1\right)}{2R^{2}}=0,\\\
\frac{q_{x}q_{y}^{2}\sec(\theta)\left(-3\sec(\theta)\left(2R^{2}(\sin(2\theta)c_{21}-2\tan(\theta)c_{03})+\cos(2\theta)-1\right)-8R^{3}c_{22}\right)}{4R^{2}}=0,\\\
-\frac{q_{x}q_{y}\left(2R^{2}c_{21}+\tan(\theta)\right)}{R}=0,\\\
-\frac{q_{x}^{2}q_{y}\sec(\theta)\left(5\sin(\theta)\left(2R^{2}c_{21}+\tan(\theta)\right)+2R^{2}(3\tan(\theta)\sec(\theta)c_{03}+2Rc_{22})\right)}{2R^{2}}=0,\\\
-\frac{q_{x}^{2}\left(2R^{2}c_{21}+\tan(\theta)\right)}{2R}=0,\\\
\frac{q_{y}^{3}\sec^{3}(\theta)\left(-32R^{2}(2\sin(\theta)c_{03}+Rc_{04})-3\cos(\theta)+3\cos(3\theta)\right)}{8R^{2}}=0,\\\
-\frac{3q_{y}^{2}\left(2R^{2}\sec^{2}(\theta)c_{03}+\tan(\theta)\right)}{2R}=0.\end{dcases}$
(5.5)

The Eqs. (5.5) are solved using exact arithmetic and give results for the
surface expansion coefficients shown in Table 4, which agree exactly with
those given in [5].

$c_{mn}$ | Values
---|---
$c_{21}$ | $2.53388\times 10^{-6}$
$c_{03}$ | $2.43386\times 10^{-6}$
$c_{40}$ | $-6.55111\times 10^{-10}$
$c_{22}$ | $-3.77553\times 10^{-9}$
$c_{04}$ | $-3.02209\times 10^{-9}$
Table 4: Surface expansion coefficients for the second example defined
according to Eq. (4.3).

The last example we present in this paper is a comparison between spot
diagrams of a biconic mirror when computed using OpticStudio and our Lie
method; see Figure 6. The mapping in Eq. (4.11) generates a third-degree
polynomial in phase-space variables which can be used as a ray-tracer between
object and image plane coordinates. In Figure 7, we can see the difference in
the ray-tracing for an off-axis beam of rays originating from the object point
at position $\bm{q}=(-0.5,0.5)$ and direction domain
$\bm{p}\in[-0.0075,0.0125]\times[-0.0125,0.0075]$ and for an on-axis beam of
rays originating from $\bm{q}=(0,0)$ with direction domain
$\bm{p}\in[-0.01,0.01]^{2}$. For both cases the object and image distances are
$s_{\mathrm{ob}}=200,\,s_{\mathrm{im}}=100$ and the OAR has an incidence angle
equal to $\theta=\pi/6$. The surface equation of the biconic is:

$z=\zeta(\bm{q})=\frac{c_{x}q_{x}^{2}+c_{y}q_{y}^{2}}{1+\sqrt{1-(1+\kappa_{x})c_{x}^{2}q_{x}^{2}-(1+\kappa_{y})c_{y}^{2}q_{y}^{2}}},$
(5.6)

with $\kappa_{x}=\kappa_{y}=0$ and $c_{x}=-\dfrac{\sqrt{3}}{200}$,
$c_{y}=-\dfrac{3\sqrt{3}}{800}$. Using the surface expansion coefficients of
Eq. (5.6) we can determine the necessary coefficients $b_{ijkl}$ for the Lie
operators and the resulting spot diagram coincides almost perfectly with the
OpticStudio ray tracing. The maximum distance between the coordinates given by
the two methods in the examples is $\Delta_{\mathrm{max}}=9\times 10^{-5}$.

Figure 6: Sketch of the biconic reflector in Eq. (5.6). The point objects $1$
and $2$ are imaged paraxially onto their primed counterparts. Figure 7: Spot
diagrams at image plane. Left: Off-axis object point. Right: On-axis object
point.

##  6 Conclusions

In this paper we extend the procedure presented for rotationally symmetric
systems in [12, 13, 9] to mirror systems with only planar symmetry. Starting
from a set of analytical ray-tracing equations, we expand them up to third-
order. The information about these expansions is then encoded into the
associated Lie transformations. We derive the generator polynomials for the
Lie transformations up to fourth degree. Thus, the method produces third-order
analytical expressions for the transverse ray aberrations for an arbitrary
mirror with planar symmetry. We calculate coefficients of the generators for a
single mirror. These coefficients depend only on the geometrical information
of the mirror itself. It is therefore possible to describe an arbitrary
optical system as the concatenation of single mirrors since for each mirror
the associated generated polynomials are known. Complex phenomena like lower
order aberrations combining into higher order ones are captured by the method.

We verified our results with three applications. In the first two, we show how
it is possible to use the analytic expressions of the aberrations to determine
the freeform coefficients of the mirror surface that eliminate aberrations up
to third-order in the case of a point object and an object at infinity. The
last example shows that the aberration expressions can also be used for ray
tracing (up to the order of accuracy that has been used in the Lie
transformations). Here, we see excellent agreement between the Lie-generated
spot diagrams and the ones generated by OpticStudio.

The authors now aim to explore the application of the shown method to the
limiting case of grazing incidence and to investigate possible applications
for the determination of mirror systems free of, or with reduced, third-order
aberrations. The latter can serve as advantageous starting designs for complex
mirror systems. Additionally, we intend to work out the relation between the
Lie aberration coefficients and the wavefront aberration coefficients
described in [1, 2].

##  Appendix A Additional Lie Tools

The Baker-Campbell-Hausdorff (BCH) formula describes how two Lie
transformations generated by $f$ and $g$ can be combined into a single one
generated by $k$:

$\exp([k,\cdot\,])=\exp([f,\cdot\,])\exp([g,\cdot\,]),$ (A.1a) where
$k=f+g+[f,g]/2+([f,[f,g]]+[g,[g,f]])/12+\cdots\quad.$ (A.1b)

In the current discussion, through the BCH formula Eq. (A.1) the lower degree
generators combine into higher degree contributions and therefore the
aberrations do so as well [12, 13, 9].

Furthermore, it can be proven that the following identity for a triplet of Lie
transformations holds [11]:

$\begin{gathered}\exp([g,\cdot\,])\exp([f,\cdot\,])\exp(-[g,\cdot\,])=\exp([k,\cdot\,]),\\\
k=\exp([g,\cdot\,])f.\end{gathered}$ (A.2)

These tools are important when composing Lie transformations. Consider the
example of two mirrors described by the generators $f_{2},f_{3}$ and
$g_{2},g_{3}$. To derive the ordered composition map $\mathcal{M}$ of the
combined system we proceed as follows:

$\displaystyle\mathcal{M}$
$\displaystyle=\exp([f_{2},\cdot\,])\exp([f_{3},\cdot\,])\exp([g_{2},\cdot\,])\exp([g_{3},\cdot\,])$
$\displaystyle=\exp([f_{2},\cdot\,])\underbrace{\exp([g_{2},\cdot\,])\exp(-[g_{2},\cdot\,])}_{=\mathcal{I}}\exp([f_{3},\cdot\,])\exp([g_{2},\cdot\,])\exp([g_{3},\cdot\,])$
$\displaystyle=\underbrace{\exp([f_{2},\cdot\,])\exp([g_{2},\cdot\,])}_{=\exp([k_{2},\cdot\,])}\underbrace{\exp(-[g_{2},\cdot\,])\exp([f_{3},\cdot\,])\exp([g_{2},\cdot\,])}_{=\exp([f_{3}^{\mathrm{tr}},\cdot\,])}\exp([g_{3},\cdot\,])$
$\displaystyle=\exp([k_{2},\cdot\,])\exp([f^{\mathrm{tr}}_{3},\cdot\,])\exp([g_{3},\cdot\,])$
$\displaystyle\overset{(2)}{=}\exp([k_{2},\cdot\,])\exp([k_{3},\cdot\,]),$
(A.3)

where $f^{\mathrm{tr}}_{3}=\exp(-[g_{2},\cdot\,])f_{3}$ according to Eq. (A.2)
and $k_{3}=f^{\mathrm{tr}}_{3}+g_{3}$ according to Eq. (A.1). The Lie
transformations generated by $f_{2},g_{2}$ have related matrices and the
product of these matrices determines the generator $k_{2}$. The insertion of
the identity map $\mathcal{I}$ in Eq (A.3) allows us to simultaneously reorder
the Lie transformations and apply Eq. (A.2). Note that, by using the BCH
formula there would be additional generators of degree 4 and higher, which can
be neglected if we consider ordered compositions of generators up to degree 3;
see [12, 13].

##  Appendix B Generator Coefficients

Here the reader will find the non-zero coefficients of $\tau_{4}$ in Eq.
(4.11).

Coefficients | Values
---|---
$b_{0004}$ | $-\frac{1}{8}s_{\mathrm{im}}\left(8s_{\mathrm{im}}^{2}a_{0301}+8s_{\mathrm{ob}}a_{0200}(s_{\mathrm{im}}a_{0200}+1)(2s_{\mathrm{im}}a_{0200}(s_{\mathrm{im}}a_{0200}+1)+1)-8s_{\mathrm{im}}a_{0202}+1\right)+s_{\mathrm{im}}^{4}a_{0400}-\frac{s_{\mathrm{ob}}}{8}$
$b_{0022}$ | $-\frac{1}{4}s_{\mathrm{im}}\left(4s_{\mathrm{im}}^{2}(a_{1210}+a_{2101})+4s_{\mathrm{ob}}\left(s_{\mathrm{im}}a_{2000}^{2}+s_{\mathrm{im}}a_{0200}^{2}(2s_{\mathrm{im}}a_{2000}+1)^{2}+a_{0200}(2s_{\mathrm{im}}a_{2000}+1)^{2}+a_{2000}\right)-4s_{\mathrm{im}}(a_{0220}+a_{2002})+1\right)$
|  $+s_{\mathrm{im}}^{4}a_{2200}-\frac{s_{\mathrm{ob}}}{4}$
$b_{0040}$ | $-\frac{1}{8}s_{\mathrm{im}}\left(8s_{\mathrm{im}}^{2}a_{3010}+8s_{\mathrm{ob}}a_{2000}(s_{\mathrm{im}}a_{2000}+1)(2s_{\mathrm{im}}a_{2000}(s_{\mathrm{im}}a_{2000}+1)+1)-8s_{\mathrm{im}}a_{2020}+1\right)+s_{\mathrm{im}}^{4}a_{4000}-\frac{s_{\mathrm{ob}}}{8}$
$b_{0103}$ | $s_{\mathrm{ob}}a_{0200}(2s_{\mathrm{im}}a_{0200}+1)^{3}+s_{\mathrm{im}}(s_{\mathrm{im}}(3a_{0301}-4s_{\mathrm{im}}a_{0400})-2a_{0202})$
$b_{0121}$ | $s_{\mathrm{ob}}a_{0200}(2s_{\mathrm{im}}a_{0200}+1)(2s_{\mathrm{im}}a_{2000}+1)^{2}+s_{\mathrm{im}}(s_{\mathrm{im}}(-2s_{\mathrm{im}}a_{2200}+2a_{1210}+a_{2101})-2a_{0220})$
$b_{0202}$ | $6s_{\mathrm{im}}^{2}a_{0400}-3s_{\mathrm{ob}}a_{0200}^{2}(2s_{\mathrm{im}}a_{0200}+1)^{2}-3s_{\mathrm{im}}a_{0301}+a_{0202}$
$b_{0220}$ | $-s_{\mathrm{ob}}a_{0200}^{2}(2s_{\mathrm{im}}a_{2000}+1)^{2}+s_{\mathrm{im}}(s_{\mathrm{im}}a_{2200}-a_{1210})+a_{0220}$
$b_{0301}$ | $4s_{\mathrm{ob}}a_{0200}^{3}(2s_{\mathrm{im}}a_{0200}+1)-4s_{\mathrm{im}}a_{0400}+a_{0301}$
$b_{0400}$ | $a_{0400}-2s_{\mathrm{ob}}a_{0200}^{4}$
$b_{1012}$ | $s_{\mathrm{im}}\left(s_{\mathrm{im}}^{2}\left(8s_{\mathrm{ob}}a_{0200}^{2}a_{2000}^{2}-2a_{2200}\right)+s_{\mathrm{im}}(4s_{\mathrm{ob}}a_{0200}a_{2000}(a_{0200}+2a_{2000})+a_{1210}+2a_{2101})+2s_{\mathrm{ob}}a_{2000}(2a_{0200}+a_{2000})-2a_{2002}\right)+s_{\mathrm{ob}}a_{2000}$
$b_{1030}$ | $s_{\mathrm{ob}}a_{2000}(2s_{\mathrm{im}}a_{2000}+1)^{3}+s_{\mathrm{im}}(s_{\mathrm{im}}(3a_{3010}-4s_{\mathrm{im}}a_{4000})-2a_{2020})$
$b_{1111}$ | $-2(s_{\mathrm{im}}(4s_{\mathrm{ob}}a_{0200}a_{2000}(a_{2000}(2s_{\mathrm{im}}a_{0200}+1)+a_{0200})-2s_{\mathrm{im}}a_{2200}+a_{1210}+a_{2101})+2s_{\mathrm{ob}}a_{0200}a_{2000})$
$b_{1210}$ | $4s_{\mathrm{ob}}a_{2000}a_{0200}^{2}(2s_{\mathrm{im}}a_{2000}+1)-2s_{\mathrm{im}}a_{2200}+a_{1210}$
$b_{2002}$ | $-s_{\mathrm{ob}}a_{2000}^{2}(2s_{\mathrm{im}}a_{0200}+1)^{2}+s_{\mathrm{im}}(s_{\mathrm{im}}a_{2200}-a_{2101})+a_{2002}$
$b_{2020}$ | $6s_{\mathrm{im}}^{2}a_{4000}-3s_{\mathrm{ob}}a_{2000}^{2}(2s_{\mathrm{im}}a_{2000}+1)^{2}-3s_{\mathrm{im}}a_{3010}+a_{2020}$
$b_{2101}$ | $4s_{\mathrm{ob}}a_{0200}a_{2000}^{2}(2s_{\mathrm{im}}a_{0200}+1)-2s_{\mathrm{im}}a_{2200}+a_{2101}$
$b_{2200}$ | $a_{2200}-4s_{\mathrm{ob}}a_{0200}^{2}a_{2000}^{2}$
$b_{3010}$ | $4s_{\mathrm{ob}}a_{2000}^{3}(2s_{\mathrm{im}}a_{2000}+1)-4s_{\mathrm{im}}a_{4000}+a_{3010}$
$b_{4000}$ | $a_{4000}-2s_{\mathrm{ob}}a_{2000}^{4}$

Funding TKI program “Photolitho M&CS" (TKI-HTSM 19.0162)

Acknowledgements The authors thank Teus Tukker (ASML) for his fruitful
remarks.

Disclosures The authors declare no conflicts of interest.

Data availabilityData underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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