# Bounds on Power and Common Message Fractions for RSMA with Imperfect SIC

Garima Chopra, Akhileswar Chowdary, Abhinav Kumar, and Marwa Chafii Garima
Chopra is with Chitkara University Institute of Engineering and Technology,
Chitkara University, Punjab, India-173212, Akhileswar Chowdary is with NYU
WIRELESS, New York University Tandon School of Engineering, Brooklyn, New
York, Abhinav Kumar is with the Department of Electrical Engineering, Indian
Institute of Technology Hyderabad, Telangana, India-502285, and Marwa Chafii
is with the Engineering Division, New York University (NYU) Abu Dhabi, Abu
Dhabi and NYU WIRELESS, NYU Tandon School of Engineering, Brooklyn, New York
(Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>marwa.chafii@nyu.edu). This work is supported in
part by the SERB NPDF–project: PDF/2020/001251 and DST NMICPS through TiHAN
Faculty fellowship of Dr. Abhinav Kumar.

###### Abstract

Rate-Splitting multiple access (RSMA) has emerged as a key enabler in
improving the performance of the beyond fifth-generation (5G) cellular
networks. The existing literature has typically considered the sum rate of the
users to evaluate the performance of RSMA. However, it has been shown in the
existing works that maximizing the sum rate can result in asymmetric user
performance. It significantly enhances one user’s rate at the cost of the rate
of another RSMA user. Further, imperfections can reduce the performance of
successive interference cancellation (SIC)-based RSMA. Therefore, in this
letter, we consider the imperfection in SIC and derive suitable bounds on
fractions of the power allocated for common and private messages and the
fraction of common message intended for each user in an RSMA pair such that
their individual RSMA rates are greater than their respective orthogonal
multiple access (OMA) rates. Through simulations, we validate the derived
bounds. We show that they can be used to appropriately select the RSMA
parameters resulting in users’ RSMA rates being better than their respective
OMA rates.

###### Index Terms:

Beyond Fifth Generation (5G) cellular networks, Rate-Splitting Multiple Access
(RSMA), Successive Interference Cancellation (SIC).

##  I Introduction

Rate splitting multiple access (RSMA) is gaining popularity due to its
robustness, reliability, and high throughput in comparison to other multiple
access schemes: orthogonal multiple access (OMA), non-orthogonal multiple
access (NOMA) [1]. RSMA is an established multiple access scheme for which
NOMA and space division multiple access (SDMA) act as special cases. For a
downlink RSMA, the user’s message is split into common and private messages at
the transmitter, and the successive interference cancellation (SIC) is
executed at the receiver’s side [1].

In [2], the authors have worked on the sum-rate maximization for wireless
networks that use RSMA in the downlink. It has been shown that RSMA can
outperform orthogonal frequency division multiple access (OFDMA) and NOMA in
terms of data rate with $23.5\%$ and $19.6\%$ gains, respectively. The authors
in [3] have developed a novel transmission scheme that utilizes RSMA with
artificial noise and adaptive beamforming to maximize the secrecy sum rate. In
[4], the authors have applied RSMA for a multi-group and multi-cast scenario
where each message is intended for a group of users. They have also designed
the physical layer and performed a link-level simulation for RSMA and SDMA by
assuming cellular and multi-beam satellite systems. The comparative analysis
in [5] shows that RSMA is more robust and efficient than NOMA for multiple-
input multiple-output (MIMO) settings. An extensive study on key multiple
access technologies for aerial networks has been presented in [6]. The authors
in [6] have modeled and analyzed the weighted sum-rate performance of two user
networks served by an RSMA-based aerial base station (BS). The authors in [7]
have derived an optimal power allocation strategy and demonstrated that RSMA
with the proposed strategy is robust to degrading performance due to user
mobility compared to the conventional MIMO strategies.

Most of the existing works on RSMA have concentrated on the sum-rate
maximization and other aspects related to the sum rate of the users. However,
a study on the individual user rates in RSMA is required. An asymmetric
increase in user rates can result in unfairness between RSMA users. Motivated
by this, we derive the bounds on the power allocation of common and private
messages and the fraction of the individual common message for which the RSMA
rates of strong and weak users will be greater than their respective OMA rates
simultaneously. To the best of our knowledge, this is the first work that
derives the bounds on power allocation and common message fraction by
considering the imperfection in SIC for RSMA. Therefore, by fixing the power
allocation fractions of common and their individual private messages and the
fraction of common message within the derived bounds, individual rates of RSMA
users will always be greater than their respective OMA rates. The main
contributions of the letter are as follows.

  * •

We derive bounds on power allocation coefficients of common and private
messages and the fraction of individual common messages for RSMA for a two-
user scenario with imperfect SIC.

  * •

Through extensive simulations, we show that the bounds are close to the
numerically obtained values.

The organization of the letter is as follows. In Section II, the system model
is presented. The bounds on the fraction of common message for a user and
power allocation coefficients for common and private messages are derived in
Section III. The derived bounds are validated in Section IV. The conclusion
and future work are presented in Section V.

##  II System Model

We consider a downlink RSMA system for a BS as depicted in Fig. 1. Let
$\pazocal{U}=\\{1,2,3,...,N\\}$ be the set of users associated with the BS $b$
under consideration for a given user association scheme and $N$ be the total
number of users connected with BS $b$. In downlink transmission, the signal-
to-interference-plus-noise ratio (SINR) from BS $b$ to user $u$
($u\in\pazocal{U}$) on a subchannel $m$ in case of OMA is given as

$\gamma_{u}=\frac{P_{t}\|h_{u}\|^{2}}{\sigma^{2}+I_{u}},$ (1)

where $P_{t}$ is the maximum transmit power of BS $b$, $\|h_{u}\|^{2}$ is the
channel gain between user $u$ and BS $b$, and $\sigma^{2}$ is the noise
variance. $I_{u}=\sum_{\hat{b}\in\mathcal{B},\hat{b}\neq
b}P_{t}^{\hat{b}}\|h_{u}\|^{2}$ is the total interference from neighboring
BSs, where $P_{t}^{\hat{b}}$ and $\mathcal{B}$ is the transmit power of
neighboring BSs and set of BSs in a given area, respectively. The downlink
rate for user $u$ based on the log-rate model is given as [1]

$R^{\text{oma}}_{u}=\frac{1}{2}\log_{2}(1+\gamma_{u}),$ (2)

Figure 1: System Model.

We consider a 1-layer RSMA system for a pair of RSMA users, i.e., $s$ and $w$
$(s\;\text{and}\;w\in\pazocal{U})$. The proposed bounds can be implemented for
all the RSMA pairs. Without loss of generality, we assume that
$\|h_{s}\|^{2}>\|h_{w}\|^{2}$. Therefore, we denote $s$ as the strong user and
$w$ as the weak user. The users’ messages, $s$ and $w$ are divided into their
respective common and private messages. Using RSMA, different power levels are
assigned to common and private messages of strong and weak users. Let $P_{c}$,
$P_{ps}$, and $P_{pw}$ be the power of common stream of $s$ and $w$, private
stream of $s$, and private stream of $w$, respectively, such that
$P_{c}+P_{ps}+P_{pw}\leq P_{t}$.

The receiver of the respective users performs SIC to decode their messages. At
the receiver side, the user $s$ decodes the common message by treating the
interference from the private messages as noise. Then, the user $s$ subtracts
the common message from the received signal to extract its own private message
by treating other users’ private messages as noise. Therefore, the SINR of the
common message of user $s$ is given as follows.

$\gamma_{cs}=\frac{P_{c}\|h_{s}\|^{2}}{(P_{ps}+P_{pw})\|h_{s}\|^{2}+\sigma^{2}+I_{u}}.$
(3)

Dividing numerator and denominator of (3) by $(\sigma^{2}+I_{u})$ and
replacing $\|h_{s}\|^{2}/(\sigma^{2}+I_{u})$ with $\gamma_{s}/P_{t}$ from (1),
we get

$\gamma_{cs}=\frac{P_{c}\gamma_{s}}{(P_{t}-P_{c})\gamma_{s}+P_{t}}.$ (4)

Additionally, the SINR of the private message of user $s$ is

$\gamma_{ps}=\frac{P_{ps}\gamma_{s}}{\beta
P_{c}\gamma_{s}+P_{pw}\gamma_{s}+P_{t}},$ (5)

where $P_{c}=\alpha_{c}P_{t}$, $P_{ps}=\lambda(P_{t}-P_{c})$,
$P_{pw}=(1-\lambda)(P_{t}-P_{c})$, $\alpha_{c}\in(0,1)$ is the power fraction
of common message, $\lambda\in(0,1)$ is the power fraction of the private
message of $s$, and $\beta\in[0,1]$ represents the coefficient of SIC
imperfection. Similarly, for user $w$, the SINRs of common and private
messages are, respectively,

$\gamma_{cw}=\frac{P_{c}\gamma_{w}}{(P_{t}-P_{c})\gamma_{w}+P_{t}},$ (6)
$\gamma_{pw}=\frac{P_{pw}\gamma_{w}}{\beta
P_{c}\gamma_{w}+P_{ps}\gamma_{w}+P_{t}}.$ (7)

Therefore, the RSMA rates for user $s$ and $w$ based on the SINRs obtained in
(4), (5), (6), and (7) are

$R^{\text{rsma}}_{s}=\tau R^{\text{comm.}}+R^{\text{priv.}}_{s},$ (8)
$R^{\text{rsma}}_{w}=(1-\tau)R^{\text{comm.}}+R^{\text{priv.}}_{w},$ (9)

where $R^{\text{comm.}}_{x}=\log_{2}(1+\gamma_{cx})$,
$R^{\text{priv.}}_{x}=\log_{2}(1+\gamma_{px})$, $x\in\\{s,w\\}$, and
$R^{\text{comm.}}=\min(R^{\text{comm.}}_{s},R^{\text{comm.}}_{w})$ [1].
$\tau\in(0,1)$ is the fraction of common message intended only for user $s$
from $R^{\text{comm.}}$. From the expression of $R^{\text{comm.}}$, the common
rate relies on the common message rate of the weak user, i.e., $w$ [1]. Hence,
we take $R^{\text{comm.}}\approx R^{\text{comm.}}_{w}$ for evaluating the
bounds on the power fractions in the following section. The sum-rate for an
RSMA pair is computed using (8) and (9) as

$SR^{\text{rsma}}=R^{\text{comm.}}+R^{\text{priv.}}_{s}+R^{\text{priv.}}_{w}.$
(10)

Using (2), the OMA sum-rate for the same pair is given as

$SR^{\text{oma}}=R^{\text{oma}}_{s}+R^{\text{oma}}_{w}.$ (11)

##  III Bounds on Common Message fraction of $s$ ($\tau$), Power Fraction of
Common Message ($\alpha_{c}$), and Power Fraction of Private Message of $s$
($\lambda$)

In this section, we derive bounds on $\tau$, $\alpha_{c}$, and $\lambda$ for
which the individual RSMA rates of user $s$ and $w$ ($R^{\text{rsma}}_{s}$ and
$R^{\text{rsma}}_{w}$) should be greater than their respective OMA rates
($R^{\text{oma}}_{s}$ and $R^{\text{oma}}_{w}$).

###  III-A Bounds on $\tau$

We consider that $R^{\text{rsma}}_{s}>R^{\text{oma}}_{s}$ and
$R^{\text{rsma}}_{w}>R^{\text{oma}}_{w}$ for user $s$ and $w$, respectively.
Using these constraints and (2), (8), (9), we derive the upper and lower
bounds on $\tau$. The constraints are expressed as follows

$\tau R^{\text{comm.}}+R^{\text{priv.}}_{s}>R^{\text{oma}}_{s}.$ (12)

Substituting $R^{\text{comm.}}$, $R^{\text{priv.}}_{s}$, and
$R^{\text{oma}}_{s}$ with their respective expressions from (6), (5), and (2)
in (12), we obtain

$\tau>\frac{1}{\log_{2}(1+\gamma_{cw})}\bigg{[}\frac{1}{2}\log_{2}(1+\gamma_{s})-\log_{2}(1+\gamma_{ps})\bigg{]}.$
(13)

Substituting $\gamma_{cw}$ and $\gamma_{ps}$ from (6) and (5) in (13) and
further solving (13), we get

$\tau>\frac{\log_{2}\bigg{(}\frac{\sqrt{(1+\gamma_{s})}(\beta\alpha_{c}\gamma_{s}+\gamma_{s}(1-\lambda)(1-\alpha_{c})+1)}{\alpha_{c}\gamma_{s}(\beta-1)+(\gamma_{s}+1)}\bigg{)}}{\log_{2}\big{(}\frac{(\gamma_{w}+1)}{\gamma_{w}(1-\alpha_{c})+1}\big{)}}.$
(14)

Similarly, the constraint for user $w$ is given as follows

$(1-\tau)R^{\text{comm.}}+R^{\text{priv.}}_{w}>R^{\text{oma}}_{w}.$ (15)

Using $\gamma_{cw}$ and $\gamma_{pw}$ from (6) and (7) in (15), we get

$\tau<\frac{\log_{2}\bigg{(}\frac{(\gamma_{w}+1)(\alpha_{c}\gamma_{w}(\beta-1)+(\gamma_{w}+1))}{(\sqrt{1+\gamma_{w}})(\gamma_{w}(1-\alpha_{c})+1)(\beta\alpha_{c}\gamma_{w}+\lambda\gamma_{w}(1-\alpha_{c})+1)}\bigg{)}}{\log_{2}\bigg{(}\frac{(\gamma_{w}+1)}{\gamma_{w}(1-\alpha_{c})+1}\bigg{)}}.$
(16)

The bounds derived in (14) and (16), respectively, are considered as lower and
upper bound for $\tau$ and referred to as $\tau_{\text{lower}}$ and
$\tau_{\text{upper}}$, respectively. Note that the upper and lower bounds on
$\tau$ depend on $\alpha_{c}$, $\beta$, and $\lambda$.

###  III-B Bounds on $\alpha_{c}$

The range of $\tau$ is $(0,1)$. Rearranging (12), we get

$\tau>\frac{R^{\text{oma}}_{s}-R^{\text{priv.}}_{s}}{R^{\text{comm.}}}.$ (17)

Considering $\tau>0$ in (17), we obtain,

$\frac{R^{\text{oma}}_{s}-R^{\text{priv.}}_{s}}{R^{\text{comm.}}}>0\implies
R^{\text{oma}}_{s}>R^{\text{priv.}}_{s},$
$\frac{1}{2}\log_{2}(1+\gamma_{s})>\log_{2}(1+\gamma_{ps})\implies\gamma_{ps}<\sqrt{1+\gamma_{s}}-1.$
(18)

Substituting the expression of $\gamma_{ps}$ from (5) in (18), and solving
further, we get

$\alpha_{c}>\frac{\sqrt{1+\gamma_{s}}(\lambda\gamma_{s}-\gamma_{s}-1)+\gamma_{s}+1}{\gamma_{s}(\sqrt{1+\gamma_{s}}(\beta-1+\lambda)+1-\beta)}\triangleq\alpha_{\text{LB}}.$
(19)

The $\alpha_{c}$ should satisfy (19) for a given $\lambda$ and $\beta$ for
$\tau_{\text{lower}}$ to be greater than 0. Similarly, considering and
rearranging (15), we get,

$\tau<1-\bigg{(}\frac{R^{\text{oma}}_{w}-R^{\text{priv.}}_{w}}{R^{\text{comm.}}}\bigg{)}.$
(20)

Solving (20) by imposing the constraint that the right-hand side (RHS) of the
expression should be less than 1, we get,

$\frac{R^{\text{oma}}_{w}-R^{\text{priv.}}_{w}}{R^{\text{comm.}}}>0\implies
R^{\text{oma}}_{w}>R^{\text{priv.}}_{w},$ $\gamma_{pw}<\sqrt{1+\gamma_{w}}-1.$
(21)

Replacing $\gamma_{pw}$ with (7) in (21) and further solving, we get

$\alpha_{c}<\frac{\sqrt{1+\gamma_{w}}(\lambda\gamma_{w}+1)-(1+\gamma_{w})}{\gamma_{w}(\sqrt{1+\gamma_{w}}(\lambda-\beta)+\beta-1)}.$
(22)

Along with (19), the $\alpha_{c}$ should also satisfy (22) for a given
$\lambda$ and $\beta$ for $\tau_{\text{upper}}$ to be less than 1. We also
need to select an $\alpha_{c}$ such that
$\tau_{\text{lower}}<\tau_{\text{upper}}$. Hence, solving the inequality
$(\ref{tau_lb})<(\ref{tau_ub})$, we get the following cubic equation in
$\alpha_{c}$.

$-\alpha_{c}^{3}\gamma_{s}\gamma_{w}^{2}B+\alpha_{c}^{2}\Bigg{[}\gamma_{w}\bigg{(}\gamma_{s}B(\gamma_{w}+1)-C\bigg{)}\\\
-\frac{\gamma_{s}\gamma_{w}(\beta-1)^{2}(\gamma_{w}+1)}{A}\Bigg{]}+\alpha_{c}\Bigg{[}\gamma_{w}(C-D)+C-\\\
\frac{(\beta-1)(\gamma_{w}+1)(\gamma_{s}-\gamma_{w})}{A}\Bigg{]}+\Bigg{[}(\gamma_{w}+1)\\\
\left(D-\frac{(\gamma_{s}+1)(\gamma_{w}+1)}{A}\right)\Bigg{]}<0,$ (23)

where

$\displaystyle A$ $\displaystyle=\sqrt{1+\gamma_{s}}\sqrt{1+\gamma_{w}},$
$\displaystyle B$ $\displaystyle=\beta(\beta-1)+\lambda(1-\lambda),$
$\displaystyle C$
$\displaystyle=\beta(\gamma_{s}+\gamma_{w})+\gamma_{s}\gamma_{w}(\beta-\lambda(1-\lambda))$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\lambda(\gamma_{s}\gamma_{w}(1-\lambda)+\gamma_{w}-\gamma_{s})-\gamma_{s},$
$\displaystyle D$
$\displaystyle=\lambda(\gamma_{s}\gamma_{w}(1-\lambda)+\gamma_{w}-\gamma_{s})+\gamma_{s}+1.$

The $\alpha_{c}$ should satisfy (19), (22), and (23) for a given $\lambda$ and
$\beta$. The expressions in (19), (22), and (23) are functions of $\lambda$
and $\beta$. Thus, $\lambda$ and $\beta$ have a significant impact on the
value of $\alpha_{c}$. Therefore, we need to choose a $\lambda$ such that the
bounds of $\alpha_{c}$ always lie between 0 and 1. Considering this condition,
we next derive bounds on $\lambda$.

###  III-C Bounds on $\lambda$

We consider the constraint that $\alpha_{c}$ lies between $(0,1)$. Applying
this constraint on the bounds of $\alpha_{c}$ derived in (19) and (22), we
derive the bounds on $\lambda$. Considering numerator of (19) to be greater
than $0$, and further solving, we get

$\lambda>\frac{(\sqrt{1+\gamma_{s}}-1)\sqrt{1+\gamma_{s}}}{\gamma_{s}}.$ (24)

Now, applying the positivity constraint on the denominator of (19), we obtain,

$\lambda>\frac{(\sqrt{1+\gamma_{s}}-1)(1-\beta)}{\sqrt{1+\gamma_{s}}}.$ (25)

By applying the constraint on (19) that $\alpha_{c}<1$, we get

$\frac{\sqrt{1+\gamma_{s}}(\lambda\gamma_{s}-\gamma_{s}-1)+\gamma_{s}+1}{\gamma_{s}(\sqrt{1+\gamma_{s}}(\beta-1+\lambda)+1-\beta)}<1,$
(26)

Considering that (25) is true, we obtain,

$\beta>\frac{-1}{\gamma_{s}}.$ (27)

We already know that $\beta\in[0,1]$. Hence, $\beta$ always satisfies (27).
This implies that the lower bound on $\alpha_{c}$ in (19) is always less than
1. Similarly, applying the positivity constraint on the numerator and
denominator of (22), we get

Figure 2: Variation of the cubic equation and region of operation of
$\alpha_{c}$ with respect to (w.r.t.) $\lambda$. Figure 3: Variation of the
suitable upper bound (23) and lower bounds of $\alpha_{c}$ (19) w.r.t.
$\lambda$.

$\displaystyle\lambda$
$\displaystyle>\frac{\sqrt{1+\gamma_{w}}-1}{\gamma_{w}},$ (28)
$\displaystyle\lambda$
$\displaystyle>\frac{\beta(\sqrt{1+\gamma_{w}}-1)+1}{\sqrt{1+\gamma_{w}}}.$
(29)

By setting (22) less than $1$ and assuming (29) to be true, we get

$\beta<\frac{-1}{\gamma_{w}}.$ (30)

The condition obtained in (30) cannot be true as $\beta\in[0,1]$. Hence, the
upper bound on $\alpha_{c}$ in (22) cannot be less than $1$. Therefore, it is
not a strict upper bound on $\alpha_{c}$. Further, we have also derived a few
other bounds on $\lambda$ by applying the constraint that the numerator and
denominator of (19) and (22) are less than 0. However, we found that those
bounds are insignificant. Due to space constraints, we are not presenting them
in this letter.

###  III-D Strict lower bound on $\lambda$ and Strict upper bound on
$\alpha_{c}$

We have plotted the left-hand side (LHS) of (23) for different values of
$\lambda$ and $\alpha_{c}$ in Fig. 2 for $\gamma_{s}=6$ dB and $\gamma_{w}=2$
dB. Using these SINR values, we have computed the lower bound of $\lambda$
using (24), (25), (28), and (29) which is approximately equal to $0.7$. Hence,
we have plotted the LHS of (23) for various $\lambda$ starting from $0.7$. The
values of $\alpha_{\text{LB}}$ for different $\lambda$ are evaluated using
(19). The vertical lines in Fig. 2 represent the lower bounds of $\alpha_{c}$
for different $\lambda$. The inequality shown in (23) states that the cubic
expression in $\alpha_{c}$ on the LHS should be less than $0$. Therefore, for
a particular $\lambda$, we need to select $\alpha_{c}$ such that it satisfies
$\alpha_{c}>\alpha_{\text{LB}}$ and the inequality mentioned in (23). From
Fig. 2, it is observed that for the lower bound on $\lambda$ computed using
(24), (25), (28), and (29) no $\alpha_{c}$ satisfies (23). As we increase
$\lambda$, we observe that for $\lambda\geq 0.865$ there exists $\alpha_{c}$
which satisfies $\alpha_{c}>\alpha_{\text{LB}}$ and (23). Therefore, the lower
bound of $\lambda$ obtained using Section III-C is not a strict lower bound.
The suitable lower bound of $\lambda$ is the lowest value of $\lambda$ for
which there exists a solution of (23) which is denoted as
$\lambda_{\text{lower}}$. In addition, (23) provides the suitable upper bound
of $\alpha_{c}$ denoted by $\alpha_{\text{UB}}$ which is the point where the
curve cuts the x-axis after the vertical line. Similarly, we have plotted the
$\alpha_{\text{LB}}$ and $\alpha_{\text{UB}}$ for $\lambda\geq 0.865$
(obtained from Fig. 2). Since the bounds of $\alpha_{c}$ depend on $\lambda$
as shown in (19) and (23), we have plotted variation of $\alpha_{c}$ w.r.t
$\lambda\geq\lambda_{LB}$ in Fig. 3. It is observed that as the value of
$\lambda$ increases, the region of operation of $\alpha_{c}$ widens.
Summarizing Fig. 2 and Fig. 3 for a given $\gamma_{s}$ and $\gamma_{w}$, there
exists a strict lower bound on $\lambda$ and upper bound on $\alpha_{c}$ whose
closed-form expression can be derived using (23). We have further validated
these bounds in the following section. The computation of the RSMA rates of
the strong and weak users using the derived bounds is presented in Algorithm
1.

Algorithm 1 Steps for calculating RSMA rates of strong and weak users using
the derived bounds.

1: INPUTS: $\gamma_{s}$ and $\gamma_{w}$.

2: OUTPUTS: $\tau$, $\alpha_{c}$, $\lambda$, $R^{\text{rsma}}_{s}$, and
$R^{\text{rsma}}_{w}$ .

3: For a given $\gamma_{s}$ and $\gamma_{w}$, compute $\lambda_{\text{lower}}$
using (23).

4: Select $\lambda>\lambda_{\text{lower}}$.

5: Using the selected $\lambda$, compute $\alpha_{\text{LB}}$ and
$\alpha_{\text{UB}}$ using (19) and (23), respectively.

6: Select an $\alpha_{c}$ such that
$\alpha_{\text{LB}}<\alpha_{c}<\alpha_{\text{UB}}$.

7: Using the selected $\alpha_{c}$ and $\lambda$, compute
$\tau_{\text{lower}}$ and $\tau_{\text{upper}}$ using (14) and (16),
respectively.

8: Select a $\tau$ such that $\tau_{\text{lower}}<\tau<\tau_{\text{upper}}$.

9: Using the selected $\tau$, $\alpha_{c}$, and $\lambda$, compute
$R^{\text{rsma}}_{s}$ and $R^{\text{rsma}}_{w}$ as in (8) and (9),
respectively.

Figure 4: Variation of the rates of $s$ and $w$ w.r.t $\lambda$. Figure 5:
Variation of the rates of $s$ and $w$ w.r.t $\alpha_{c}$.

##  IV Results and Discussions

In this section, we validate the aforementioned bounds through simulation. We
consider a two users scenario with interference as in [2]. Fig. 4 illustrates
the variation of RSMA and OMA rates (in bps/Hz) of $s$ and $w$ w.r.t.
$\lambda$. For $\gamma_{s}=6$ dB, $\gamma_{w}=2$ dB, $\beta=0$ (i.e., for
perfect SIC), $P_{t}=1$ W, we have observed that with increase in $\lambda$,
the $SR^{\text{rsma}}$ increases beyond $SR^{\text{oma}}$ after
$\lambda=0.865$. The $\tau$ values computed using $\lambda<0.865$ are
negative, which is inappropriate for computing the rates. Hence, the RSMA
rates in the region, $\lambda<0.865$, are indicated as zero. The value of
$\lambda=0.865$ is greater than the value derived using (24), (25), (28), and
(29). This shows that the bound obtained using (24), (25), (28), and (29) is
not strict. Thus, the lower bound of $\lambda$ computed using (23) and
presented in Fig. 2 is validated in Fig. 4. Nevertheless, selecting the
$\lambda$ using (24), (25), (28), and (29) is necessary to make the bound of
$\alpha_{c}>0$. Furthermore, it is observed from Fig. 3 for
$\lambda>\lambda_{\text{lower}}$ that $R^{\text{rsma}}_{s}$ and
$R^{\text{rsma}}_{w}$ are greater than $R^{\text{oma}}_{s}$ and
$R^{\text{oma}}_{w}$, respectively. Note that the private message of user $s$
requires more power than that of user $w$.

Fig. 5 shows the variation of rate w.r.t $\alpha_{c}$ for $\beta=0$ for the
same set of $\gamma_{s}$ and $\gamma_{w}$. For $\lambda=0.99$, the lower and
upper bound of $\alpha_{c}$ computed using (19) and (23), respectively, are
$0.683$ and $0.776$. It is evident from the Fig. 5 that for
$\alpha_{\text{LB}}<\alpha_{c}<\alpha_{\text{UB}}$, the individual RSMA rates
of $s$ and $w$ are greater than their respective OMA rates. Moreover, the
lower and upper bound of $\alpha_{c}$ computed using (19) and (23) are
approximately equal to the $\alpha_{\text{LB}}$ and $\alpha_{\text{UB}}$
observed in Fig. 5. Furthermore, similar results are obtained by varying
$\tau$ for $\lambda=0.99$ and $\alpha_{c}=0.689$. However, due to space
constraints, we cannot present the plots.

Figure 6: Variation of $\tau_{\text{lower}}$ and $\tau_{\text{upper}}$ w.r.t
$\alpha_{c}$ for $\lambda=0.99$. Figure 7: Variation of the rates of $s$ and
$w$ w.r.t $\beta$.

Since the derived bounds of $\tau$ as shown in (14) and (16) are functions of
$\alpha_{c}$ and $\lambda$, we have plotted the variation of the bounds w.r.t
$\alpha_{c}$ for $\lambda=0.99$ in Fig. 6. $\alpha_{\text{LB}}$ and
$\alpha_{\text{UB}}$ are computed using (19) and (23), respectively. For
$\beta=0$, $\gamma_{s}=6$ dB, and $\gamma_{w}=2$ dB, $\tau_{\text{lower}}$ and
$\tau_{\text{upper}}$ are plotted by varying $\alpha_{c}$ between
$\alpha_{\text{LB}}$ and $\alpha_{\text{UB}}$. We can observe that these
bounds are almost equal to the theoretical bounds computed using (14) and
(16). Similar results can be obtained for different $\lambda$.

Fig. 7 shows the variation of rate w.r.t. $\beta$. We can observe that as
$\beta$ increases, after 0.035, the $R^{\text{rsma}}_{s}$ becomes less than
$R^{\text{oma}}_{s}$. However, the $SR^{\text{rsma}}$ remains greater than
$SR^{\text{oma}}$ as $\beta$ increases further. Nevertheless, after
$\beta=0.05$, $SR^{\text{rsma}}$ starts deteriorating. This is due to the
decrease in $\gamma_{ps}$ and $\gamma_{pw}$ with an increase in $\beta$. The
result in Fig. 7 is plotted for $\lambda=0.99$, $\alpha_{c}=0.689$, and
$\tau=0.1$.

##  V Conclusion

In this letter, we have derived the bounds on the power allocation
coefficients of the common and private messages and the fraction of the
individual common message of the users for which the RSMA rates of users will
be greater than their respective OMA rates. We have validated the derived
bounds using simulation. We have numerically computed $\alpha_{c}$, $\lambda$,
and $\tau$, which are validated using simulation. In our future work, we aim
to compute the closed-form expressions of the lower bound of $\lambda$ and
$\alpha_{\text{UB}}$. We also aim to derive bounds on $\beta$ in our future
work.

## References

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# Size and Shape Fluctuations of Ultrasoft Colloids

Huarui Wu  Department of Engineering Physics and Key Laboratory of Particle
and Radiation Imaging (Tsinghua University) of Ministry of Education, Tsinghua
University, Beijing 100084, China.  Jing Song  Department of Engineering
Physics and Key Laboratory of Particle and Radiation Imaging (Tsinghua
University) of Ministry of Education, Tsinghua University, Beijing 100084,
China.  Wei-Ren Chen  Neutron Scattering Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee 37831, United States  Kun Song  Department of
Engineering Physics and Key Laboratory of Particle and Radiation Imaging
(Tsinghua University) of Ministry of Education, Tsinghua University, Beijing
100084, China.  Lionel Porcar  Institut Laue-Langevin, B.P. 156, F-38042
Grenoble CEDEX 9, France  Zhe Wang<EMAIL_ADDRESS>Department
of Engineering Physics and Key Laboratory of Particle and Radiation Imaging
(Tsinghua University) of Ministry of Education, Tsinghua University, Beijing
100084, China.

###### Abstract

Ultrasoft colloidal particle fluctuates due to its flexibility. Such
fluctuation is essential for colloidal structure and dynamics, but is
challenging to quantify experimentally. We use dendrimers as a model system to
study the fluctuation of ultrasoft colloids. By considering the dynamic
polydispersity in the small-angle neutron scattering (SANS) model, and
introducing the fluctuation of invasive water into the contrast in SANS, we
reveal the fluctuating amplitudes of the size and shape of the dendrimer of
generation 6 at finite concentrations. The size fluctuation is suppressed
while the shape fluctuation increases as the weight fraction of dendrimers
passes 11%. With neutron spin echo data, we suggest that such crossover
originates from the competition between the inter- and intra-particle
dynamics. Further investigation on lower-generation samples shows a contrary
result, which suggests a structural basis for these dynamic phenomena.

Ultrasoft colloids, such as dendrimers and star polymers, are featured by
extraordinary molecular flexibility [1, 2, 3, 4, 5, 6]. The elastic energy
stored by such a particle that undergoes a large strain can be just hundreds
of or even tens of the thermal energy [7, 8, 9, 10], which distinguishes them
from emulsions, most microgels, or other common deformable particles [11, 12,
13, 14, 10, 15, 16, 17]. Consequently, ultrasoft colloids exhibit significant
molecular fluctuations. These fluctuations are crucial in many physical
processes. Simulations suggest that the size and shape fluctuations
considerably affect the self-diffusion of particles [18]. At concentrations
close to the random close packing, it is proven that the shape fluctuation is
related to the stress-releasing and -building of particle, and plays an
essential role in the unusual dynamics [19]. On the practical level,
fluctuations can modulate the particle conformation [20] that profoundly
impacts a large variety of applications [21, 22, 23, 24]. For instance,
dendrimers have been established as drug carriers. The conformational
fluctuation directly affects the size, shape and internal cavity of the
molecule, which are important for the drug loading and release ability and the
permeability across the biomembrane [25, 26, 27, 28, 29]. Therefore, there is
a strong need to quantify the size and shape fluctuations to advance our
understanding on both the physics and applications of ultrasoft colloids.

In the past two decades, neutron spin echo (NSE) technique has been used to
measure the intra-particle motions of ultrasoft colloids at the dilute limit,
and the fluctuations of the structural unit and shape of the isolated particle
were revealed [30, 31]. While many ultrasoft colloidal suspensions of physical
and technological importance are with finite concentrations [10, 19, 32]. For
these systems, the NSE analysis on the molecular fluctuation is much
complicated by the difficulty in experimentally determining the collective
translational diffusion of particles and its coupling to the intra-particle
dynamics [33]. On the other hand, the fluctuations influence the distribution
of the particle conformation, which could be reflected in small-angle
scattering (SAS) patterns. Note that, most previous SAS studies of ultrasoft
colloids have not explicitly considered the effect of particle size and shape
fluctuations [1, 34]. Many SAS analyses deteriorate at volume fractions higher
than about 10%. For example, the calculated SAS curve may not well match the
position or height of the main peak [7, 9], or underestimate the intensity at
small $Q$ ($\bm{Q}$ is the scattering vector) [35]. Thus, it is possible that
the fluctuation effect is important in interpreting the SAS data of ultrasoft
colloids, and can be extracted by reasonable modelling.

In this letter, we investigate the size and shape fluctuations of ultrasoft
colloids by using neutral poly(amido amine) (PAMAM) dendrimers dissolved in
water as the model system. Six concentrations, $c=1$, 5, 10, 12.5, 15, and 20
wt% (weight fraction of dendrimers) were measured. We first introduce an
approach to examine these fluctuations in the dendrimer molecule by small-
angle neutron scattering (SANS). It will be seen that explicitly considering
the size fluctuation is of special importance in SANS analysis. Then, by
combining the NSE result, and investigating lower-generation samples, we
reveal the mechanism for the dependences of these fluctuations on the
molecular structure and dynamics.

Dendrimers with the same molecular weight possess different conformations due
to molecular fluctuations. Thus, dendrimer solutions can be regarded as
polydisperse. We introduce the dynamic polydispersity [36] to reflect this
fluctuation effect, which leads to the following expression of the SANS
intensity [37]:

$I(Q)=n_{\textrm{p}}AP(Q)S^{\prime}(Q),$ (1)

where $n_{\textrm{p}}$ is the number density of dendrimer molecules, $A$
denotes the contrast of the scattering length between solute particle and
solvent, $P(Q)$ is the average form factor normalized at $Q=0$, and
$S^{\prime}(Q)$ is the apparent structure factor given by
$S^{\prime}(Q)=1+\beta(Q)[S(Q)-1]$, where $\beta(Q)$ is the polydispersity
factor [37] that incorporates the size and shape fluctuations into the
analysis, and $S(Q)$ is the inter-particle structure factor. $S(Q)$ is
calculated by the Percus-Yevick closure of the Ornstein-Zernike equation [38]
with a Gaussian pair potential [9].

We first explore the size fluctuation by modeling the fluctuating dendrimers
as a collection of polydisperse spheres. So, $P(Q)$ is expressed as [36, 39]:

$\displaystyle
P_{\textrm{s}}(Q)=\sqrt{\frac{2}{\pi}}\frac{1}{\sigma_{R}}\left[1+\textrm{erf}\left(\frac{R}{\sqrt{2}\sigma_{R}}\right)\right]^{-1}$
$\displaystyle\times\int_{0}^{\infty}\left[\frac{3j_{1}(Qr)}{Qr}\right]^{2}\textrm{exp}\left[-\frac{(r-R)^{2}}{2\sigma_{R}^{2}}\right]dr+a_{\textrm{b}}P_{\textrm{b}}(Q),$
(2)

where $j_{1}(x)$ is the 1st order spherical Bessel function of the 1st kind,
$P_{\rm b}(Q)$ represents the contribution from the intra-particle density
variation with $a_{\rm b}$ denoting its amplitude [36, 40, 41], $R$ is the
average radius, and $\sigma_{R}$ is the standard deviation of the Gaussian
distribution of $R$. $\sigma_{R}$ reflects the fluctuating amplitude of the
dendrimer size. This form factor ignores the “fuzzy” profile of the radial
density distribution [36, 5]. It could be reasonable for high-generation
dendrimers, because their ratios of the fuzzy edge ($\sigma_{\rm f}$, see Eq.
(3)) to radius are relatively small [36]. The analyses for the samples of
generation 6 (G6) are displayed in Fig. 1 (a, blue lines), and are seen to be
satisfactory at the measured concentrations.

To highlight the effect of the size fluctuation, we analyze the same data with
the frequently-used monodisperse fuzzy-ball model [36, 42, 43]. Here, the size
fluctuation is not considered, so $\beta(Q)=1$ and $S^{\prime}(Q)=S(Q)$.
$P(Q)$ is given by [36]:

$P_{\textrm{f}}(Q)=\left[\frac{3j_{1}(QR)}{QR}\textrm{exp}\left(-\frac{Q^{2}\sigma_{\textrm{f}}^{2}}{4}\right)\right]^{2}+a_{\textrm{b}}P_{\textrm{b}}(Q),$
(3)

where $\sigma_{\textrm{f}}$ denotes the spatial range of the fuzzy edge. This
model has the same number of parameters as the preceding one. The fitting
results, shown in Fig. 1(b), exhibit clear discrepancies from the measured
spectra at $c\geq 10$ wt%. The low-$Q$ intensity is remarkably underestimated.
The peak position also deviates from the measured one. Comparing the two
models, we assert that the size fluctuation is significant, and should be
taken into account in analyzing the SAS data at $c\geq 10$ wt%. Numerically,
the incorporation of the size fluctuation lowers the low-$Q$ part of
$\beta(Q)$ from 1 since $\beta(Q=0)=\langle r^{3}\rangle^{2}/\langle
r^{6}\rangle$ [37], and thus lifts the low-$Q$ part of $S^{\prime}(Q)$ to
match the experimental data [44]. Notice that, at $c\leq 5$ wt%, both models
work well, leading to contradictory results on the existence of the size
fluctuation. This issue will be clarified in the scattering contrast analysis
shown later.

Figure 1: SANS analyses on the G6 PAMAM dendrimers dissolved in $\rm{D_{2}O}$
at $c=1$, 5, 10, 12.5, 15 and 20 wt%. Symbols in (a) and (b) denote the
measured spectra(they are vertically shifted for visibility). (a) Fits with
the size fluctuation model and the model with both the size and shape
fluctuations. (b) Fits with the monodisperse fuzzy-ball model and the model
only considering the shape fluctuation. These two models, without explicitly
considering the size fluctuation, underestimate the low-$Q$ intensity at
$c\geq 10$ wt%. (c) Size fluctuation ($\sigma_{R}$) of the G6 dendrimer as a
function of $c$. The results are obtained with the size fluctuation model and
the model with both the size and shape fluctuations. (d) Shape fluctuation
($\sigma_{\epsilon}$) of the G6 dendrimer as a function of $c$. The details of
the SANS fitting can be found in the Supplementary Material.

Considering that the dendrimer can deform to a spheroid-like form [45], we
incorporate the shape fluctuation by allowing the aspect ratio of a dendrimer
($\epsilon$) to fluctuate according to a Schultz distribution [37] with the
center at 1 and the standard deviation of $\sigma_{\epsilon}$. The fitting
results considering both the size and shape fluctuations are given in Fig.
1(a, magenta lines). It is seen that incorporating the shape fluctuation
improves the fit at high $Q$. Notice that, solely incorporating the shape
fluctuation cannot match the low-$Q$ spectra (Fig. 1(b)), since in this case
$\beta(Q\to 0)$ does not deviate from 1 [44].

Figure 1(c) and (d) show the fitting results for $\sigma_{R}$ and
$\sigma_{\epsilon}$ of the G6 dendrimer, respectively. At the dilute limit,
$\sigma_{R}$ is 4.7 $\mathrm{\mathring{A}}$, corresponding to a dynamic
polydispersity $\xi_{R}=\sigma_{R}/R$ of 15%. As $c$ crosses about 11 wt%, the
size fluctuation is suppressed while the shape fluctuation is strongly
enhanced. In later part we will show that this behavior corresponds to a
dynamic crossover.

As seen from Fig. 1, incorporating the size fluctuation is important in
decoding the structural information of ultrasoft colloids. Next, we will
verify its existence from another view. Since dendrimer has a water-accessible
architecture [46, 47], the contrast term $A$ in Eq. (1) is written as
$A=\langle(b_{\rm{pol}}+Nb_{\rm{w}}-n_{\rm{w}}V_{\rm{p}}b_{\rm{w}})^{2}\rangle$,
where $b_{\rm{pol}}$ is the total scattering length of a dry dendrimer,
$b_{\rm{w}}$ is the average scattering length of a water molecule, $N$ is the
number of water molecules inside a dendrimer, $V_{\rm{p}}$ is the volume of a
dendrimer in solution, $n_{\rm{w}}$ is the number density of bulk water, and
$\langle\dots\rangle$ denotes the average over all particles. It is
straightforward to find that:

$A(b_{\rm w})=[n_{\rm{w}}^{2}(\langle V_{\rm p}\rangle^{2}+\langle\Delta
V_{\rm p}^{2}\rangle)+\langle N\rangle^{2}+\langle\Delta
N^{2}\rangle-2n_{\rm{w}}(\langle V_{\rm p}\rangle\langle
N\rangle+\langle\Delta V_{\rm p}\Delta N\rangle)]b_{\rm
w}^{2}-2b_{\rm{pol}}(n_{\rm w}\langle V_{\rm p}\rangle-\langle N\rangle)b_{\rm
w}+b_{\rm pol}^{2},$ (4)

where $\Delta V_{\rm p}=V_{\rm p}-\langle V_{\rm p}\rangle$ and $\Delta
N=N-\langle N\rangle$. $\Delta N$ and $\Delta V_{\rm p}$ should be highly
correlated. To the first-order approximation, it can be assumed that $\Delta
N$ is proportional to $\Delta V_{\rm p}$ by $\Delta N=n_{\rm{in}}\Delta V_{\rm
p}$, where $n_{\rm{in}}$ should be smaller than $n_{\rm{w}}$ due to the
excluded volume of the constituent atoms of dendrimer. So that Eq. (4) is
rewritten as:

$A(b_{\rm w})=(\alpha^{2}+\theta^{2})b_{\rm w}^{2}-2b_{\rm pol}\theta b_{\rm
w}+b_{\rm pol}^{2},$ (5)

where $\theta=n_{\rm w}\langle V_{\rm p}\rangle-\langle N\rangle$, and
$\alpha=(n_{\rm w}/n_{\rm in}-1)\sigma_{N}$ with
$\sigma_{N}=\sqrt{\langle\Delta N^{2}\rangle}$ denoting the fluctuation of the
number of invasive water molecules. If no such fluctuation exists, then
$\alpha=0$, and Eq. (5) reduces to:

$A(b_{\rm w})=(\theta b_{\rm w}-b_{\rm pol})^{2}.$ (6)

Equations (5) and (6) provide an approach to verify the existence of the size
fluctuation ($\propto\sigma_{N}$) from a microscopic view. One can vary
$b_{\rm w}$ by changing the molar fraction of $\rm{D_{2}O}$ in solvent
($\gamma$), and fit the experimental $A(b_{\rm w})$ with Eqs.(5) and (6). If
the size fluctuation is considerable, the fitting quality with Eq. (5) will be
better than that with Eq. (6). We vary $\gamma$ from 100% to 60% for all
concentrations, and fit the experimental contrast term with Eqs.(5) and (6).
The results are shown in Fig. 2. It is seen that at $c\leq 10$ wt%, Eq. (5)
performs much better than Eq. (6), suggesting that the fluctuations of size
and invasive water exist and strongly affect the scattering contrast. At
$c>10$ wt%, this effect is less significant, implying a smaller fluctuation.
The values of $\alpha/N$, representing the fluctuation of invasive water, are
given in Fig. 3(a). In the same panel, we also plot the particle volume
fluctuation $\sigma_{V}/V_{\rm p}$ obtained from the model fitting of SANS
spectra as illustrated in Fig. 1(a). It is remarkable that these two
quantities display very similar behaviors, especially considering that they
are found with different approaches [48]. Such consistency confirms the
existence of the size fluctuation in the G6 dendrimer.

Figure 2: Scattering contrast $A$ as a function of the average scattering
length of solvent molecule $b_{\rm w}$ at $c\leq 10$ wt% (a) and $c>10$ wt%
(b). Symbols denote the experimental results. Solid and dashed lines denote
the fits using Eq. (5) (with size fluctuation) and Eq. (6) (without size
fluctuation), respectively [48]. They are shifted vertically for visibility.
Figure 3: (a) Fluctuations of volume ($\sigma_{V}/V_{\rm p}$) and invasive
water ($\alpha/N\propto\sigma_{N}/N$) of a G6 dendrimer as a function of $c$.
(b) Inter-particle collisional time $\uptau_{\rm{inter}}$ and intra-particle
relaxation time $\uptau_{\rm{intra}}$ of G6 dendrimers as a function of $c$
[49]. The dashed line marks the crossover concentration $c^{*}$.

As mentioned above, SAS analyses on neutral ultrasoft colloids usually
deteriorate when $c$ is larger than about 10 wt%. It has been tentatively
attributed to the failure of factorizing $I(Q)$ into the product of $P(Q)$ and
$S(Q)$ due to the inter-particle interpenetration or overlap [50, 7, 5].
Nevertheless, many studies show that strong interpenetration or overlap does
not occur at such low concentrations [51, 52, 53, 54, 55, 56, 57]. Our
analysis suggests that the failure of $P(Q)\cdot S(Q)$ factorization is due to
the molecular fluctuation, and can be corrected with Eq. (1). In fact,
Pedersen also found that the SANS intensity of block copolymer micelles is
expressed by Eq. (1) rather than a product of $P(Q)$ and $S(Q)$ by considering
the configuration distribution of the chains in corona [42]. We argue that the
particle conformational fluctuation is the basis for both Ref. [42] and our
work. This agreement suggests that Eq. (1) is the better expression to account
for the softness, which originates from the flexible architecture, of
ultrasoft particles such as dendrimers and starlike polymers. Note that, Eq.
(1) is based on the assumption that the particle conformation decouples with
the position [37]. It becomes invalid at concentrations close to random close
packing because of the considerable spatial heterogeneity of particle
deformation [19].

As seen from Fig. 1 and Fig. 3(a), all discussed fluctuations exhibit
crossovers at $c^{*}\approx 11$ wt%. Here, we seek for the dynamic origin of
this phenomenon. Figure 3(b) shows the results of an NSE study on the same
system [49]. By decomposing the motion of a dendrimer into the translational
diffusion and the internal relaxation, this NSE analysis gives two
characteristic times: the inter-particle collisional time $\uptau_{\rm inter}$
and the intra-particle relaxation time $\uptau_{\rm intra}$. Interestingly,
$\uptau_{\rm inter}$ and $\uptau_{\rm intra}$ also intersect at $c^{*}$. So,
we associate the fluctuation crossover with the dynamic process. At $c<c^{*}$,
conformational fluctuations emerge due to the enormous internal degrees of
freedom. The inter-particle collisions do not strongly perturb the internal
relaxation since $\uptau_{\rm intra}<\uptau_{\rm inter}$. While at $c>c^{*}$,
$\uptau_{\rm intra}>\uptau_{\rm inter}$, the frequent collisions hinder the
dendrimer from fully exploring its conformational space at the dilute limit,
and thus restrain the size fluctuation. Meanwhile, these collisions enhance
the particle deformation and suppress the internal relaxation, which results
in an increasing shape fluctuation.

Figure 4: (a) SANS analysis of G4 PAMAM dendrimers dissolved in $\rm D_{2}O$
at $c=1$, 5, 10, 12.5, 15 and 20 wt%. Symbols denote the measured spectra.
Solid and dashed lines respectively denote the fits using the monodisperse
fuzzy-ball model and the size fluctuation model. (b) Scattering contrast of G4
dendrimers as a function of $b_{\rm{w}}$. Symbols denote the experimental
results. Solid and dashed lines respectively denote the fits using Eq. (5) and
Eq. (6). Data are vertically shifted.

We further measure the samples of generation 4 (G4) to investigate the
structural origin of the observed fluctuations. The polydisperse sphere model
and monodisperse fuzzy-ball model are applied to fit the SANS spectra and the
results are shown in Fig. 4(a). In contrast to the G6 samples, for G4 samples
the fuzzy-ball model works well, while the polydisperse sphere model
overestimates the low-$Q$ intensity, and gives an inappropriate oscillation at
high $Q$. This result indicates that the size fluctuation is imperceptible for
G4 samples. To confirm this observation, we perform the contrast analysis
given by Eqs.(4)-(6). As seen from Fig. 4(b), Eqs.(5) and (6) give the same
fits, showing that the size fluctuation term $\alpha$ equals to zero. The
sharp difference between the G4 and G6 samples reveals the role of molecular
structure in determining the fluctuations. For our G4 sample, the ratio of the
fuzzy edge to radius ($\sigma_{\textrm{f}}/R$) is 74%, which demonstrates the
pronouncing open feature of its periphery. Intuitively, the strong fuzzyness
implies an unsharp boundary, which makes the particle volume and its
fluctuation not so well-defined. On the contrary, G6 dendrimers are found to
be alike spheres with a clearer boundary. This result agrees with the previous
finding that the dendrimer structure evolves from an open one to a compact one
as the generation increases [45, 46, 36, 58, 59]. The evident compactness in
G6 dendrimers leads to small intra-particle free volume, which enhances the
interaction between dendrons [60, 55, 5]. Therefore, it is reasonable that the
dendrons can fluctuate collectively with long-range correlation, and form the
size fluctuation of the whole particle. For G4 dendrimers, a simulation shows
that the internal fluctuations are uncorrelated at large distances [20] due to
the open structure. In this case, intra-particle motions involving long-range
correlated motion of monomers, such as the size fluctuation, should be weak.
In addition, an NSE study suggests that the molecular fluctuation of low-
generation starlike dendrimers is dominated by the breathing mode that does
not induce significant size and shape fluctuations [31]. This feature is also
attributed to the fuzzy nature of the radial density profile of the molecule
[31].

In summary, we investigate the fluctuation of ultrasoft colloids by using
PAMAM dendrimers as the model system. By considering the dynamic
polydispersity and the fluctuation of invasive water in SANS analysis, we
reveal the fluctuating amplitudes of the size and shape of the G6 dendrimer.
The size fluctuation is found to be of particular importance in interpreting
the SAS data. These fluctuations exhibit strong dependences on the dynamics
and structure. The competition between the intra- and inter-particle dynamics
introduces a crossover concentration above which the size fluctuation is
suppressed, and the shape fluctuation increases. The absence of size
fluctuation in G4 dendrimers highlights the importance of the structural
compactness in determining the intra-particle motion. Our approach provides a
basis for further exploration on the fluctuation effect on various structural
and dynamic properties of ultrasoft colloids.

###### Acknowledgements.

This research was supported by the National Natural Science Foundation of
China (No. 11975136, U1830205). We thank the EQ-SANS at Spallation Neutron
Source, Oak Ridge National Lab and the D22 beamline at Institut Laue-Langevin
for beamtime.

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HRR source of  duration  pulses
M. Ehret, J. Cikhardt, P. Bradford, et al.

High-repetition-rate source of nanosecond duration
-current pulses driven by relativistic laser pulses

1]Michael EhretC Adaja 8, ES-37185, or<EMAIL_ADDRESS>
2]Jakub Cikhardt

3]Philip Bradford

1]Iuliana-Mariana Vladisavlevici

4]Tomas Burian

1]Diego de Luis

1]Jose Luis Henares

1]Rubén Hernández Martín

1]Jon Imanol Apiñaniz

1]Roberto Lera

1]José Antonio Pérez-Hernández

3]João Jorge Santos

1]Giancarlo Gatti

[1]Centro de Láseres Pulsados (CLPU), Villamayor, Spain
[2]Czech Technical University in Prague, Faculty of Electrical Engineering, Prague, Czech Republic
[3]Univ. Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications (CELIA), UMR 5107, Talence, France
[4]Department of Radiation and Chemical Physics, FZU-Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic

We report the first high-repetition rate generation and simultaneous characterization of nanosecond-scale return currents of -magnitude issued by the polarization of a target irradiated with a PW-class high repetition rate Ti:Sa laser system at relativistic intensities. We present experimental results obtained with the VEGA-3 laser at intensities from 5e181.3e20□. A non-invasive inductive return current monitor is adopted to measure the derivative of return-currents on the order of  and analysis methodology is developed to derive return-currents. We compare the current for copper, aluminium and Kapton targets at different laser energies. The data shows the stable production of current peaks and clear prospects for the tailoring of the pulse shape, promising for future applications in high energy density science, e.g. electromagnetic interference stress tests, high-voltage pulse response measurements, and charged particle beam lensing. We compare the target discharge of the order of hundreds of  with theoretical predictions and a good agreement is found.

§ INTRODUCTION

The continuous technical and scientific improvement of lasers [1, 2] has led to stable short-pulse PW-class high repetition rate Ti:Sa systems [3, 4]. If these lasers are tightly focused onto matter, the relativistic interaction yields forward-acceleration of electrons [5] that in turn can trigger pulsed bright ion beams by well known mechanisms such as Target Normal Sheath Acceleration (TNSA) [6, 7], Radiation Pressure Acceleration (RPA) [8] and others [9] beneficial to isotope production [10], positron emission tomography [11], ion beam microscopy [12], Particle-Induced X-ray Emission (PIXE) [13] as well as inertial confinement fusion [14]. The mechanisms rely on the build up of large accelerating potentials which are also the source of ultra-strong electromagnetic pulses [15]. In particular, targets attain a strong positive net-charge due to laser-accelerated electrons that are able to escape the rising potential barrier [16]. As a result of this, -level discharge pulses and return currents can be produced and propagate over the target surface [17]. Interest in these effects is twofold: (i) both are sensitive to the total amount of charge that leaves the target and therefore can be used as a passive diagnostic of the laser-target interaction; and (ii) both allow to deliver all-optically generated -level -duration current pulses that can be understood as olivea novel secondary source.

The monitoring of target discharge is an important aspect of ultrahigh intensity laser-solid interaction at high-repetition-rate. This paper presents an inductive current monitor as metrology for high-voltage pulses driven at high-repetition-rate. The measurement of return currents with inductive current monitors has been demonstrated previously in the regime of -driver lasers with intensities 1e141e16□ [30]. We apply this technique to the characterization of discharge pulses driven by high power Ti:Sa systems at relativistic intensities.

Pulses of -level at -duration pose a risk for electronic systems in the vicinity of the interaction [18, 19], but they also have an application in the context of proton beam focusing [20, 21] oliveand transient magnetic field generation [17]. We demonstrate here the stable generation of discharge pulses, with a clear perspective to obtain a novel high-repetition-rate source of -scale current pulses for future applications, , e.g. on the field of electromagnetic compatibility (EMC) tests  [22], radio-location [23], military technologies [24], biology [25], and medicine [26].

§ MATERIALS AND METHODS

The primary diagnostic used in this study was a Target Charging Monitor (TCM) constructed based on the principles of an inductive current monitor [30]. The TCM measures the derivative of the current that streams through the device as shown in Fig. <ref>. Key advantage of this metrology technique is its destruction-free nature. Current pulses are excited by laser-plasma interaction on a solid density target, transported through the TCM and can be applied after their characterization.

Cut of the Target Charging Monitor (TCM) with two opposite MHV-BNC connectors with soldered pins to pass through the pulsed current issued by relativistic laser interaction in the top to the application side in the bottom. The TCM comprises a solid copper body forming a cup; with a cylindrical top; both later of which are separated by dielectric material POM. The through current induces a magnetic field enclosed in the cylinder which causes an induced current to flow in a small squared loop formed by the core of a RG142 coaxial cable connected to an output SMA connector. The current pulse itself is issued by the discharge of a solid tape target (top) and coupled into oliveone of the support rods of the tape which are connected to a RG142 coaxial cable leading to the TCM. oliveThe other support rod is isolated from the ground.

Current pulses that pass through the TCM device induce a magnetic field inside the cylindrical copper body which causes an induced current to flow in a coil-shaped rod connected to an coaxial output. The calibration factor which relates the time-integrated voltage to the current is -2.0 ±0.3 e9 (see Appx. <ref>). For this work, current pulses are transported via RG142 coaxial cables and the circuit impedance is $Z=\SI{50}{\ohm}$. The through signal is terminated in the facility grounding. Note that cable lengths are measured with 3 FWHM voltage pulses: target and TCM are connected with a coaxial cable of 9.6 ±0.2 length; and TCM and grounding are connected with a coaxial cable of 13.6 ±0.1 length. Induced signals are transported to a 2 oscilloscope and acquisitions are throughout corrected for the frequency dependent attenuation of circuit elements. Circuit calibrations are done using a R&S ZNH 4 vector network analyzer. The effective bandwidth of the circuit is 2.

Experiments for this work are conducted in the VEGA-3 laser facility at CLPU with high-power Ti:Sa laser pulses amplified to an energy $E_\mathrm{L}$ up to 25 per pulse measured behind the compressor. After compression to a duration $\tau_\mathrm{L}$ of 30, the short laser pulse is transported in high-vacuum of 1e-6 via a $f=$ 2.5 off-axis parabola onto the target with a beam-transport efficiency of 82. The focal spot of $d_\mathrm{L}~=~\SI{12.8 \pm 1.9}{\micro\metre}$ full-width at half-maximum (FWHM) is maintained at constant size. The energy on target is extrapolated from calibrations recorded at low-energy and the focal spot at high energy is estimated to be the same as for low-energy measurements. Note the large Rayleigh length of 160 ±50, for this work with a non-difraction limited focal spot. For this experiment, 33.9 ±1.5 of the energy on target are within in the first Airy disk at low power. In addition there are three hot spots aside the focus in the first Airy ring, containing in total as much as 20 ±2 of the energy on target with an average intensity of 24 ±2 of the main intensity. The pulse duration is measured on-shot with a second-harmonic autocorrelator system that diagnoses the faint reflection from a thin pellicle positioned between parabola and focus.

The tape target system TaTaS-PW [27] transported aluminium tape of 10 ±1 thickness, Kapton tape of 89 ±9 thickness, tape of 10 ±1 aluminium enforced with Kapton (Al-e-K) [27] and copper tape of 7 ±1 thickness across the laser focal plane. oliveTapes are 12.5 wide stripes. The conductive 5 diameter support metal rods which guide the tape are 16 separated. Solid-density targets are placed in the laser focus position and tilted by 12.5 with respect to the laser axis to avoid reflection back towards laser amplifiers. As the VEGA laser pulse shows no pre-pulses capable of inducing a transparency or breakdown of the target [28], the main acceleration mechanism of charged particles is TNSA with the deployed laser and target parameters. In TNSA, a population of laser-heated oliveforward-directed relativistic electrons escapes the target oliveafter crossing its thickness and the successive potential dynamics and electron-refluxing leads to the formation of sheath fields which are capable of accelerating ionized surface contaminants up to several tens of  [29]. The TCM measures the return current towards the target and allows us to deduce the total target charging by relativistic electrons. Note that the tapeolive's only connection to the ground is one of the support metal rods. The other metal rod is floating with a total conductive length of 95. This forces the return current to flow thought the olivenon-floating support metal rod and successively in the core of the coaxial transmission line that incorporates the TCM. The shield of the transmission line is grounded with the chassis of the tape target system.

Numerical simulations are performed to compare experimental results to theoretical predictions. The laser-driven target discharge is simulated with ChoCoLaT-2 [16] (see Appx. <ref>) - and the laser-absorption efficiency into hot electrons is studied with the particle-in-cell (PIC) code SMILEI [46] (see Appx. <ref>).

§ RESULTS AND DISCUSSION

The circuit-corrected signal of the TCM for an aluminium target exhibits a clear positive peak for the rising edge of the positive current pulse. It is preceded by a low-noise pedestal; and followed by pulses streaming from the grounding to the target: first EMP induced noise and second the reflection of the current pulse at the impedance miss-matched ground. Time-base oliveat TCM relative to laser-arrival at ≈0.

We first show results from a single, representative shot on an aluminium target to emphasize different aspects of the platform, and secondly study the effect of changes of laser and target parameters based on single-shot data and high-repetition-rate recordings.

The inductive TCM device measures the derivative $\mathrm{d}_\mathrm{t} I_\mathrm{p}$ of the pulsed current $I_\mathrm{p}$ streaming away from the target, see Fig. <ref>. Here, the laser pulse at 22.5 beam energy (after compressor) and a duration of 30.4 ±0.7 is fired onto a 10 ±1 thick aluminium target. The laser impacts on the target at zero time seen from the TCM. The spatial distance between TCM and target is 30 ±1 to ensure that the spherically expanding vacuum bound Electromagnetic Pulse (EMP) arrives at the device first. The EMP has no significant influence on the measurement as one does not notice noise in the pedestal leading to the signal. The signal exhibits a first positive peak that detects the rising edge of the current pulse streaming through the device. This indicates a positive current pulse propagating from the target to the ground. oliveWe measure the net negative charge escaping from the target – the time between electron and ion escape is too short to be resolved. The measurement shows also a reflection of the current pulse that streams back from the imperfectly impedance matched grounding towards the target; and EMP induced noise that couples into the transmission line when the spherically expanding EMP reaches the grounding.

Current pulse (blue line) from an aluminium target retrieved by numerical integration from the derivative measured with the TCM. A first short primary peak is followed by a superposition of peaks in a broad secondary peak. Time-base relative to laser-arrival at ≈0. The zero-level is controlled by comparison to a fit from before to after the current pulse (orange dashed line) – here in good agreement.

After application of the instrument calibration, the temporally-integrated signal is shown in Fig. <ref>. The peak amplitude reaches 1123 ±172. The FWHM $\tau_\mathrm{d}$ of the narrow first peak is 400. A broad second peak follows and decays slowly towards zero, which is reached after 6. The first peak corresponds to the direct coupling of the discharge pulse into the transmission line to the TCM. oliveThe difference between the shortest (direct to ground) and longest way (to opposite end of rod) from the interaction region to the exit of the grounding rod is equivalent to 100 at the speed of light. Capacitive effects may broaden the peak further. The second peak most likely comprises multiple reflections across the oliveconductive tape target.

The transported charge from an aluminium target as obtained by numerical double-integration of the derivative measured by the TCM. The integral attains a plateau only slowly due to a slightly negative tail of the return current. Time-base relative to laser-arrival at ≈0.

Further temporal integration of $Z I_\mathrm{p}^2$ yields the transported energy $E_\mathrm{p}$ and $\int I_\mathrm{p} \mathrm{d}t$ yields the transported charge $Q_\mathrm{p}$. The total transported energy is 67 ±7. The energy conversion efficiencies from laser energy on target to current pulse energy results to $\chi_T = \SI{0.4}{\percent}$; and the energy conversion efficiency only accounting for laser energy encircled in the laser focus and relativistic-intensity hot spots calculates to $\chi_S = \SI{0.6}{\percent}$. The broad second peak contains a non-negligible fraction of the pulse energy in this configuration with 43 ±7. The temporally-resolved transported charge $Q_\mathrm{p}$ is shown in Fig. <ref>. The laser extracts 2.24 ±0.34 from the target. The first peak of the current accounts for less than a third of the transported charge and the slow decay of the second peak allows the integral to reach a plateau only slowly.

For a control of the accuracy of the numerical integrations, the zero-level is compared to a fit of both plateaus before and after the current pulse, shown as orange dashed line in Fig. <ref>. Here, the zero-level is maintained.

Crucial for applications, the current pulse is reproducible over hundreds of shots oliveand is consistent with theoretical estimates. A current of 558 ±116 is obtained in $292$ shots at 1 for laser shots of 24.5 ±0.3 at 33 ±2 onto copper tape of 7 ±1 thickness. The average current and its standard deviation is shown in Fig. <ref>. The 8 stability of the current measurement indicates a good shot-to-shot stability of laser and target parameters resulting in a reproducible discharge dynamics and current production. The total transported charge amounts to 713 ±60 and the current pulse energy is 11 ±2. The energy conversion efficiencies are lower compared to the ones for aluminium: $\chi_T^\mathrm{Cu} = \SI{0.05}{\percent}$ and $\chi_S^\mathrm{Cu} = \SI{0.09}{\percent}$.

Average current and its standard deviation as obtained in $292$ laser shots of 1.0 ±0.5 e20 □ at 1 onto copper tape. Time-base relative to laser-arrival at ≈0. Multiple reflections across the conductive target yield a succession of multiple peaks.

ChoCoLaT-2 simulations predict 720 ±75 of target discharge when assuming 68 of the laser energy on target to be absorbed into electrons. Simulations take into account the experimental uncertainty for the pulse duration (33 ±2), 6.8 ±0.3 of laser energy within the first Airy disk and 4.1 ±0.3 distributed in three non-negligible hot spots with an average intensity of 24 of the main intensity. The absorption efficiency into electrons is consistent with PIC simulations, see Appx. <ref>. oliveSuch high values have been reported [31], depending on the presence of pre-plasma. If however the 2D PIC simulations should overestimate the absorption or no pre-plasma would be present, a typical [32, 33] absorption of 50 would still lead to an agreement with overlapping uncertainty intervals.

A comparison of the metallic targets above to a dielectric target reveals the likely influence of target reflections and shows how we can produce single-peak current pulses. A current of 597 ±153 is obtained in $100$ shots at 0.5 for laser shots of 22.9 ±0.2 at 33 ±1 onto Kapton tape of 89 ±9 thickness. The current oliveevolution (averaged over multiple shots) is shown in Fig. <ref>. The peak of 960 FWHM transports an average of 934 ±190 and has an energy of 13 ±4. The energy conversion efficiencies are comparable to the ones for copper: $\chi_T^\mathrm{K} = \SI{0.06}{\percent}$ and $\chi_S^\mathrm{K} = \SI{0.1}{\percent}$. In comparison to metallic targets, the primary peak is broadened due to a reduced conductivity and secondary peaks from reflections are missing. Multi-peak structures are indeed not expected to appear as reflections on tape ends do not occur, and reflections on other grounding stalks do not reach the signal transmission line with noticeable amplitude after the shot due to the low conductivity.
Fewer reflections on the target may contribute towards the reduced EMP emission that is generally observed for dielectric targets [15].

Average current and its standard deviation as obtained in $100$ laser shots of 1.0 ±0.5 e20 □ at 0.5 onto Kapton tape. The dielectric target allows to produce single pulses. Time-base relative to laser-arrival at ≈0.

The total charge measured under variation of laser pulse duration and energy as well as the target material.

It becomes clear that the current evolution and the amount of total charge varies considerably under variation of the target parameters. For a further parametric study on the variation of laser intensity, shots on aluminium tape, Kapton tape, tape of aluminium enforced with Kapton (Al-e-K) [27] and copper tape are compared in Fig. <ref>. Most charge is ejected from aluminium targets, followed by copper and Kapton. Shots on aluminium reveal a linear relation between target discharge and intensity from 2.0e191.2e20□. The platform allows for the production of tunable current pulses.

Note olivethe geometry of tape Al-e-K: two 5 wide strips of 89 thick Kapton are glued on top of the aluminium tape at both its edges on the side facing the support metal rods. This Kapton reinforcement of aluminium is at -distance from the interaction zone, so it does not change the total amount of ejected charge (consistent with the experimental measurements in Fig. <ref>). The temporal shape of the current pulse can however be influenced by the Kapton enforcement for tape Al-e-K, see Fig. <ref> (compared to Fig. <ref>). The first peak is lower, which is consistent with the reduced coupling to the grounding due to the presence of Kapton at the tape edges. As a result the secondary peaks in the tail are elevated for reasons of more charge in reflections.

Average current and its standard deviation as obtained in $25$ laser shots of 0.8 ±0.4 e20 □ at 1 onto tape Al-e-K. Time-base relative to laser-arrival at ≈0.

The characteristic parameters for all shot sequences are compared in Tab. <ref>. Current pulses from aluminium tape and reinforced aluminium tape are in a good agreement. Results for aluminium tapes exhibit higher current amplitudes when compared to copper targets due to a larger target discharge.

Comparison of current pulses from shots on aluminium tape (Al), Kapton reinforced aluminium tape (AleK), Kapton tape (Kapton) and copper tape (Cu). Laser energy measured after compressor, $N$ denotes the number of shots of the sequence, and $\chi_T$ is the ratio of energy confined in the current pulse to laser energy on target.

4claser pulse     3ctarget     4ccurrent pulse

energy     duration     rate     $N$       tape     material     thickness          peak     charge     energy     $\chi_T$

22. ±0.3     33 ±2     0.5     $87$     Al     Al     10 ±1     982 ±185     2.2 ±0.2      58 ±9     0.32

21.9 ±0.3     37 ±4     1.     25     Alek     Al     10 ±1     809 ±210     2.1 ±0.3      45 ±8     0.25

22.9 ±0.2     33 ±2     0.5     100     Kapton     Kapton     89 ±9     597 ±153     0.93 ±0.19      13 ±4     0.06

24.5 ±0.3     33 ±2     1.     292     Cu     Cu     7 ±1     558 ±116     0.71 ±0.06      11 ±2     0.05

The bandwidth of the current pulse is large and allows for applications that require broadband pulses [23], see  Fig. <ref>. Such pulses can be applied to steering antenna arrays or impulse radiating antennas to emit high power levels, i.e. in ground and subsurface radars for finding, recognition and reconstruction of moving objects.

Average power spectrum density and its standard deviation as obtained in $25$ laser shots of 0.8 ±0.4 e20 □ at 1 onto tape Al-e-K. Time-base relative to laser-arrival at ≈0.

§ CONCLUSIONS

We report the first generation and characterization of short pulsed -scale currents induced by high-power relativistic laser interaction at a high-repetition-rate. The pulses with several 100 FWHM show a < 10 stability in amplitude and a high energy conversion efficiency up to the order of 1 from laser energy to pulse energy. Optimization of the energy conversion efficiency will be possible by optimizing the target discharge based on existing theoretical models olive[16], as the experimental data appears to agree well with simulations. Current pulses can be tailored by modifying the target: experimental data shows that the return current to metallic targets is broadened due to reflections across the target, whereas the use of dielectric targets removes those reflections leading to the generation of an overall shorter pulse peak.

The highest charge of 2.2 ±0.2 is produced with aluminium targets, followed by Kapton targets with 0.93 ±0.19, and copper targets with 0.71 ±0.06.

A direct application of such pulses can be the inductive generation of pulsed strong magnetic fields in small volumes. The pulse fills a olivesolenoid if $\tau_\mathrm{d} \times c = 2\pi \times r_\mathrm{c} \times N_\mathrm{c}$, with the speed of light $c$, the radius of the coil $r_\mathrm{c}$ and $N_\mathrm{c}$ revolutions. Then the induced magnetic field in the coil centre attains $B_\mathrm{c} = \mu_0 I_\mathrm{d} c \tau_\mathrm{d} / 2\pi r_\mathrm{c} l_\mathrm{c}$, with vacuum permeability $\mu_0$ and the length of the olivesolenoid $l_\mathrm{c}$. The measured pulse of 1.1 amplitude and 400 FWHM is apt for the generation of olive11T when using 1 diameter coils of 5 length, corresponding to $40$ revolutions. Such magnetic fields can be used for the tailoring of  ions [20, 21], i.e. laser-accelerated ion beams. They are also relevant for magnetization of secondary samples [34, 35, 36] oliveif further temporally stretched, i.e. as seed fields in the context of magnetized implosions towards nuclear fusion.

Pulses of 1.1 in the $50~\Omega$ circuit correspond to pulsed voltages of 55, e.g. applicable to uni-polar nanosecond-pulse dielectric barrier discharge for producing non-thermal plasma at atmospheric pressure [37]; or the research of effects of  and sub- pulses on biological cells [38, 39, 40, 41].

§ AUTHOR CONTRIBUTIONS

The author contributions are as follows: ME, IMV performed the data acquisition, curation and analysis; ME wrote the first draft of the manuscript; JC, PB, ME, TB commissioned the device at PALS; JLH organized the beamtime at CLPU; DL, RHM managed implementation of the device;  ME, DL, JIA, RL contributed to conception and design of the study; all authors were involved with underlying experimental work; all authors contributed to manuscript improvement, read, and approved the submitted version.

§ ACKNOWLEDGEMENTS

This work would not have been possible without the help of the laser- and the engineering teams at CLPU and PALS. Special thanks to the workshops of CLPU and PALS. This work received funding from the European Union’s Horizon 2020 research and innovation program through the European IMPULSE project under grant agreement No 871161 and from LASERLAB-EUROPE V under grant agreement No 871124; and Euratom Research and Training Programme (Grant Agreements No. 633053 and No. 101052200 — EUROfusion) as well as from the Grant Agency of the Czech Republic (Grant No. GM23-05027M); and Grant PDC2021-120933-I00 funded by MCIN/ AEI / 10.13039/501100011033 and by the “European Union NextGenerationEU/PRTR”. The work was supported by funding from the Ministerio de Ciencia, Innovación y Universidades in Spain through ICTS Equipment grant No EQC2018-005230-P; further from grant PID2021-125389OA-I00 funded by MCIN / AEI / 10.13039/501100011033 / FEDER, UE and by “ERDF A way of making Europe” by the “European Union”; and in addition from grants of the Junta de Castilla y León with No CLP263P20 and No CLP087U16. oliveThe views and opinions expressed herein do not necessarily reflect those of the European Commission.

§ DATA AVAILABILITY STATEMENT

The raw data and numerical methods that support the findings of this study are available from the corresponding author upon reasonable request.

§ TCM CALIBRATION

A pulsed high voltage supply (500 FWHM) is used for the calibration of the Target Charging Monitor (TCM). The voltage supply is plugged to the top of the TCM. The through signal and the signal from the induced current are recorded on an oscilloscope of 20 bandwidth using calibrated coaxial cables in a 50 circuit, see Fig. <ref>. The current of the pulse can be naturally derived from the impedance of the circuit.

Through signal (left) and induced signal (right) corrected for attenuation of the respective circuit after the TCM.

The induced signal is integrated numerically to derive the calibration factor between pulsed current and measured current, see Fig. <ref>. The numerical integration may lead to what one observes as a change of the zero-level from before to after the peak. It is corrected for by fitting a zero-level with a linear regression. The difference between measurement and fit is fully taken into account in the following uncertainty estimates.

The integrated induced signal (orange line) shows a small offset after the pulse which might be due to numerical errors. Plateau regions before and after the peak are selected (blue dots) to fit a correction (dashed blue line) with respective uncertainty (cyan area).

The calibration factor which relates the integrated induced measurement in units of volt to the pulsed through current is obtained by fitting the base-corrected integrated induced signal to the through current, see Fig. <ref>. One obtains a calibration factor of -2.0349 ±0.3117 e9.

The integrated induced signal (blue line) is scaled to the pulsed through current (orange line) to obtain the calibration factor in units of .

§ DISCHARGE SIMULATIONS

The target discharge dynamics is studied using a detailed model of target charging in short laser pulse interactions [16] that predicts the expected discharge current due to laser-heated relativistic electrons on a thin disk target. The successive electron escape is mitigated by the target potential, based on the driver laser parameters and the interaction geometry. The model takes into account the collisional cooling of electrons within cold solid density targets. The energy and time depending hot electron distribution function $f(E,t)$ describes electrons inside the target and evolves according to

\begin{align}
\partial_t f (E,t) &= \frac{h_\mathrm{L}(E) \Theta(\tau_\mathrm{L} - t)}{\tau_\mathrm{L}}  -  \frac{f (E,t)}{\tau_\mathrm{ee}(E)} - g(E,t) \label{eq:distributionevolution}\\
h_\mathrm{L}(E) &\overset{!}{=} \frac{N_0}{T_0} \exp{\left[ - E/T_0 \right]} \\
N_0 &\overset{!}{=} \int f (E,0) \text{~d}E
\end{align}

where $h_\mathrm{L}(E)$ is a constant exponential source of hot electrons, $\Theta (t)$ the Heaviside function limiting electron heating to the laser duration, $\tau_\mathrm{ee}(E)$ the energy dependent cooling time and $g(E,t)$ the rate of electron ejection from the target. The initial hot electron temperature $T_0$ depends on laser wavelength and pulse intensity [42, 44, 43]; and $N_0$ is re-normalized to the energy balance $N_0 T_0 = \eta E_\mathrm{L}$ between the total energy of hot electrons in the target and the absorbed laser energy. Simulations require the conversion efficiency $\eta$ of laser energy to energy in the hot electron distribution, which is obtained by PIC simulations for this work (see Appx. <ref>).

The hot electron cooling time depends on target material properties such as mass density $\rho_\mathrm{t}$, mass number $A_\mathrm{t}$, atomic number $Z_\mathrm{t}$, and the hot electron
energy distribution that allows to calculate average speed $\left< v \right>_\mathrm{e}$ and energy $\left< E \right>_\mathrm{e}$. Its meticulous calculation is demonstrated in [45] with an emphasis on cases relevant for this work.

With slight modifications to the source code, we enable the addition of hot-spot driven electrons to the electron distribution function. A section is added to construct an electron distribution based on the hot-spot energy and intensity, which then is added on top of the main electron distribution function.

§ PIC SIMULATIONS

The absorption of laser pulse energy into hot electron energy is studied for the case of shots on the copper target in a typical range of pre-plasma scale lengths. The absorption results to 61 for 1 pre-plasma and 78 for 3 pre-plasma. This range covers the absorption efficiencies required to reproduce experimental target charging by discharge simulations.

The 2D PIC simulation setup consists of a solid copper target irradiated by VEGA-3 laser system under an incidence angle of 12.5. The laser is linearly polarized and has the following characteristics: a wavelength $\lambda$ of 800, a peak intensity of 7.5e19□ (corresponding to a normalized field amplitude $a_0=5.9$), a pulse duration of 33 FWHM and a transverse waist of 13. The copper target is considered fully ionized, having a density $100\rm~n_c$, and a thickness of 7. The plasma density scale length is considered 1 and 3 for the two distinct simulations, having an exponential profile over a length of 20 , and being ablated from the initial target thickness. The transverse width of the target is 40. At the rear side of the target we considered a thin layer of neutral protons of 70 thickness and $10\rm~n_c$ density to simulate target contaminants. The simulation box has 80 in the longitudinal direction and 40 in the transverse direction. The cell length is $dx=dy={\SI{12.5}{\nano\metre}}$ and the number of particles per cell is 20 for each species. The particles are deleted while crossing the domain boundaries and the fields are absorbed. The simulations were performed with Simulating Matter Irradiated by Light at Extreme Intensities (SMILEI) [46] on the cluster Supercomputación Castilla y León (SCAYLE) [47].

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