# Spot size estimation of flat-top beams in space-based gravitational wave
detectors

1,3Zhen-Xiang Hao  2Tim Haase  4,5Hong-Bo Jin  3Ya-Zheng Tao  2Gudrun
Wanner111Corresponding author<EMAIL_ADDRESS>1,3,5Ruo-Xi Wu
1,3,5Yue-Liang Wu222Corresponding author<EMAIL_ADDRESS>1Institute of
Theoretical Physics, Chinese Academy of Sciences, No.55 Zhongguancun East
Road, Haidian District, Beijing, P.R.China 100190
2Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
and Institute for Gravitational Physics of the Leibniz Universität Hannover,
Callinstr. 38, 30167 Hannover, Germany
3University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Shijingshan
District, Beijing, P.R.China 100049
4National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun
Road, Chaoyang District, Beijing, P.R.China 100101
5Hangzhou Institute for Advanced Study, University of Chinese Academy of
Sciences, Hangzhou, Zhejiang, P.R.China 310024

###### Abstract

Motivated by the necessity of a high-quality stray light control in the
detection of the gravitational waves in space, the spot size of a flat top
beam generated by the clipping of the Gaussian beam (GB) is studied. By
adopting the mode expansion method (MEM) approach to simulating the beam, a
slight variant of the definition of the mean square deviation (MSD) spot size
for the MEM beam is proposed. This enables us to quickly estimate the spot
size for arbitrary propagation distances. Given that the degree of clipping is
dependent on the power ratio within the surface of an optical element, the
power ratio within the MSD spot range is used as a measure of spot size. The
definition is then validated in the cases of simple astigmatic Gaussian beam
and nearly-Gaussian beam profiles. As a representative example, the MSD spot
size for a top-hat beam in a science interferometer in the detection of
gravitational waves in space is then simulated. As in traditional MSD spot
size analysis, the spot size is divergent when diffraction is taken into
account. A careful error analysis is carried out on the divergence and in the
present context, it is argued that this error will have little effect on our
estimation. Using the results of our study allows an optimal design of optical
systems with top-hat or other types of non-Gaussian beams. Furthermore, it
allows testing the interferometry of space-based gravitational wave detectors
for beam clipping in optical simulations. The present work will serve as a
useful guide in the future system design of the optical bench and the sizes of
the optical components.

###### keywords:

laser optical systems, space mission, gravitational wave

PACS Nos.: 42.60.-v, 07.87.+v, 04.80.Nn

##  1 Introduction

In the detection of gravitational waves in spacetime, the heterodyne
interferometry is required to track the relative displacement between two
spacecraft that are on the order of $10^{6}\
$\text{\,}\mathrm{k}\mathrm{m}\mathrm{s}$$ apart at the picometer level [1, 2,
3]. At this level of precision, many noises need to be suppressed to enhance
the signal to noise ratio of gravitational wave sources. Among these noises is
the stray light, which is generated by a number of channels, one of which is
the beam clipping when a beam of laser light impinges on optical elements of
the optical bench.

In the science phase of the mission, a Gaussian beam originating from a
spacecraft propagates to a distant spacecraft several million kilometers away
[1, 2, 3]. Upon arriving at the distant spacecraft, a small part of the
Gaussian beam wavefront is clipped and generates a top-hat beam that is
propagated into the spacecraft for interferometric measurement. Within the
optical bench, another clipping effect occurs in the sense that, on impinging
on an optical element, the finite size of the element entails that at the edge
of an element diffraction will occur and power loss of the beam is inevitable.
This in turn will generate a potential source of stray light and at the
picometer level ghost beam interference might occur and disturb the
gravitational wave measurement.

The aim of the present work is to numerically track the time evolution of the
spot size of the beam in a LISA or TAIJI type optical bench. With the stray
light problem in mind, we shall first propose to use the power ratio within
the MSD spot range as a measure of spot size. Here, the MSD spot size for the
MEM beam is a slight variant of the definition given by M. A. Porras et al[4].
After certain validations of the proposed definition, the spot size of a flat
top beam propagated within an optical bench of a LISA or TAIJI type mission is
then studied and our analysis is expected to be useful for the future system
design of the laser metrology subsystem. Common to all MSD spot size studies
of a diffracted beam, divergence inevitably occurs [5, 6, 7, 8]. A careful
error analysis is carried out to understand the impact of the divergence on
our result and it is argued that the divergence will have little effect on our
conclusion and hopefully this renders our results more trustworthy.

The paper is structured as follows. In Section 2, in terms of the power ratio
within a MSD spot range as the quality measurement of the spot size, a slight
variant of the definition of the MSD spot size of a MEM beam [4] is suggested.
In Section 3, we validate the formula for the MSD spot size of a MEM beam and
the numerical code in the cases of a simple astigmatic Gaussian beam and
Nearly-Gaussian beams which can appear in laboratory experiments. In Section
4.1, we investigate the MSD spot size in a top-hat beam case and compare this
definition with the spot size calculated from the analytic divergence angle
for the top-hat beam [9]. After that, in section Section 4.2, we calculate the
estimated value of the MSD spot size in the case of a science interferometer
in the detection of gravitational waves in space. Finally, we perform a
careful error analysis for the MSD spot size of the top-hat beam in a science
interferometer in Section 4.3.

##  2 MSD spot size for MEM beams

There are a number of spot size definitions, catering to different problems
and situations [10]. In this section, we shall put forward a definition of
spot size which is a slight variant of that given by Porras et al. [4], again
mainly motivated by the stray light problem. Further, we define the power
ratio within the surface of an optical element as a measure of the MSD spot
size.

Consider a general beam that propagates along the $z$-axis, the local electric
field can be written as

$E(x,y,z,t)=u(x,y,z)\exp{\left(i\omega t\right)},$ (1)

where $x$ and $y$ are the Cartesian coordinates in the beam cross-section,
$\omega$ is the angular frequency, $t$ is the time and $u=u(x,y,z)$ is the
complex amplitude.

Further, the MSD spot size is defined as [4]

$\begin{split}w_{j}(z)&=2\sqrt{\frac{\iint_{-\infty}^{+\infty}(j-s_{j}(z))^{2}I(x,y,z)dxdy}{P_{in}}}\\\
&=\sqrt{\frac{\iint_{-\infty}^{+\infty}j^{2}I(x,y,z)dxdy}{P_{in}}-s_{j}(z)^{2}},\end{split}$
(2)

which is two times of the square root of the variance of the intensity profile
normalized by the beam power. $j$ represents $x$ or $y$ for $x$ or $y$
direction, $P_{in}=\iint_{-\infty}^{+\infty}I(x,y,z)dxdy$ is the total power
of the beam, $I(x,y,z)=E(x,y,z)E^{*}(x,y,z)$ is the intensity of the beam for
point $(x,y,z)$ and $s_{j}(z)$ is the mean value of the transversal position
of the beam for one dimension, which is also called “energy center” [11]

$s_{j}(z)=\frac{\iint_{-\infty}^{+\infty}jI(x,y,z)dxdy}{P_{in}}.$ (3)

Based on the fact that Hermite-Gaussian (HG) modes are a complete orthonormal
set of basis functions, any electric field may be expanded in terms of an
infinite superposition of HG modes [12]. This is called the mode expansion
method (MEM). As for real applications, the infinite series needs to be
truncated at a finite number of modes as an approximation. The complex
amplitude $u=u(x,y,z)$ may be approximated as

$u(x,y,z)\approx\sum_{m=0}^{N}\sum_{n=0}^{N-m}a_{mn}u_{mn}(x,y,z).$ (4)

For notation convenience, we rewrite $u_{mn}(x,y,z)$ and $a_{mn}$ as

$u_{mn}(x,y,z)=u_{mn}=|u_{mn}|\exp{\left(i\phi_{mn}\right)},$ (5)
$a_{mn}=|a_{mn}|\exp{\left(i\beta_{mn}\right)},$ (6)

where the max mode order $N$ is the max value of the sum of mode order $m$ and
$n$, $a_{mn}=\iint_{-\infty}^{+\infty}u(x,y,z)u_{mn}^{*}(x,y,z)dxdy$ is a
constant called the complex coefficient for $u_{mn}(x,y,z)$ and
$u_{mn}(x,y,z)$ is the complex amplitude for $(m,n)$ order of symmetric HG
modes

$\begin{split}u_{mn}(x,y,z)=&\frac{c_{mn}}{w(z)}H_{m}\left(\frac{\sqrt{2}x}{w(z)}\right)H_{n}\left(\frac{\sqrt{2}y}{w(z)}\right)\exp{\left(-\frac{x^{2}+y^{2}}{w^{2}(z)}\right)}\\\
&\cdot\exp{\left(-ikz-
ik\frac{x^{2}+y^{2}}{2R(z)}+i(m+n+1)\zeta(z)\right)}.\end{split}$ (7)

For simplicity, here we use the symmetric HG modes as decomposition bases. The
symmetry here means that the beam parameters $w(x)$, $R(z)$ and $\xi(z)$ for
$x$ and $y$ directions are equal. The parameter $w(z)$ is the spot size of the
fundamental Gaussian mode $u_{00}(x,y,z)$. It is defined as
$w(z)=w_{0}\sqrt{1+\left(\frac{z}{z_{r}}\right)^{2}}$, where $w_{0}$ is the
waist, $z_{r}=\frac{\pi{w_{0}}^{2}}{\lambda}$ is the Rayleigh range, $\lambda$
is the wavelength and $k=\frac{2\pi}{\lambda}$ is the wave number. The radius
of curvature is $R(z)=z\left(1+\left(\frac{z_{r}}{z}\right)^{2}\right)$ and
the normalization constant is $c_{mn}=\left(\pi
m!n!2^{m+n-1}\right)^{-\frac{1}{2}}$. The Gouy phase $\zeta(z)$ of the
fundamental Hermite-Gaussian mode is

$\zeta(z)=\arctan\frac{z}{z_{r}}.$ (8)

The beam generated by this method is called MEM beam. In this paper, MEM
processing is performed with IfoCAD [13]. The total power $P_{MEM}$ of the MEM
beam can be written as

$P_{MEM}=\iint_{-\infty}^{+\infty}Idxdy=\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}|^{2}.$
(9)

M. A. Porras et al. calculated the MSD spot size and energy center for the
superposition of higher-order HG modes [4], but they leave out the phase term
$\phi_{mn}$ in Eq. 5 for the HG mode bases. This leads to the formulas of MSD
spot size and energy center being only valid in the waist plane of the basic
HG modes of the MEM beam. Here we add the phase term in the representation of
HG modes and substitute Eq. 4 into Eq. 3. Then, the energy center
$\left(s_{x}(z),s_{y}(z)\right)$ can be rewritten as

$\displaystyle
s_{x}(z)=\frac{w(z)}{P_{MEM}}\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{(m+1)n}|\cos{(\beta_{mn}-\beta_{(m+1)n}-\zeta(z))}\sqrt{m+1},$
(10a) $\displaystyle
s_{y}(z)=\frac{w(z)}{P_{MEM}}\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{m(n+1)}|\cos{(\beta_{mn}-\beta_{m(n+1)}-\zeta(z))}\sqrt{n+1}.$
(10b)

The MSD spot size of the MEM beam in x and y directions can be rewritten as

$\displaystyle\begin{multlined}w_{x}(z)^{2}=\frac{w(z)^{2}}{P_{MEM}}\left(\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}|^{2}(2m+1)+2\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{(m+2)n}|\right.\\\
\left.\cdot\cos{(\beta_{mn}-\beta_{(m+2)n}-2\zeta(z))}\sqrt{(m+2)(m+1)}\right)-4s_{x}(z)^{2},\end{multlined}w_{x}(z)^{2}=\frac{w(z)^{2}}{P_{MEM}}\left(\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}|^{2}(2m+1)+2\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{(m+2)n}|\right.\\\
\left.\cdot\cos{(\beta_{mn}-\beta_{(m+2)n}-2\zeta(z))}\sqrt{(m+2)(m+1)}\right)-4s_{x}(z)^{2},$
(11c)
$\displaystyle\begin{multlined}w_{y}(z)^{2}=\frac{w(z)^{2}}{P_{MEM}}\left(\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}|^{2}(2n+1)+2\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{m(n+2)}|\right.\\\
\left.\cdot\cos{(\beta_{mn}-\beta_{m(n+2)}-2\zeta(z))}\sqrt{(n+2)(n+1)}\right)-4s_{y}(z)^{2}.\end{multlined}w_{y}(z)^{2}=\frac{w(z)^{2}}{P_{MEM}}\left(\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}|^{2}(2n+1)+2\sum_{m=0}^{N}\sum_{n=0}^{N-m}|a_{mn}||a_{m(n+2)}|\right.\\\
\left.\cdot\cos{(\beta_{mn}-\beta_{m(n+2)}-2\zeta(z))}\sqrt{(n+2)(n+1)}\right)-4s_{y}(z)^{2}.$
(11f)

The above definition of spot size will be the basis for subsequent discussions
in our work. MSD spot size may be used to calculate the spot size for general
beams except for hard-edge diffraction beams [5, 4]. It has many advantages:
1. It may be used to define a good beam quality factor which characterizes the
global spatial behaviour of a laser beam [5]; 2. This definition automatically
agrees with the definition of the spot size for fundamental GB, HG beam and
Laguerre-Gaussian (LG) beam [14, 15]; 3. The MSD spot size satisfies the ABCD
law and the corresponding beam quality factor is invariable when the beam
propagates through the purely real ABCD system (the paraxial optical system
that can be described by an ABCD ray transfer matrix) [16]; 4. The MSD spot
size always is a hyperbolic function of propagation distance $z$ [17]. These
advantages in general are not shared by other spot size definitions for
general light beams [17].

A key question for applications in material processing and fibre transmission
is the amount of power inside a spot region [18]. One typical criterion of the
spot size definition is the power ratio within the spot range [14, 15]. For
detection of gravitational waves in space, we need to do the clipping test for
the optical system to probe the degree of the stray light coming from
clipping. The clipping degree is dependent on the power ratio within the
surface of optical elements. As a preliminary investigation, we calculate the
MSD spot size and the corresponding power ratio concentrated inside the MSD
spot range for the beam type which we are interested. We use the following
definition to calculate the power ratio concentrated inside the MSD spot range

$\epsilon_{P_{spot}}=\frac{\int_{s_{y}(z)-w_{y}(z)}^{s_{y}(z)+w_{y}(z)}\int_{s_{x}(z)-w_{x}(z)}^{s_{x}(z)+w_{x}(z)}Idxdy}{P_{MEM}}.$
(12)

For the Gaussian beam, HG beam and LG beam, the fractional beam energy
concentrated inside the MSD spot range is independent of $z$. As for general
beams, the power ratio within the MSD spot range is dependent on $z$ in the
near field. As for the very far-field, the power ratio within the MSD spot
range is also independent of $z$. These will be discussed in A.

The original formulas Eqs. 2 and 3 to estimate the spot size and beam center
for the beam passing through the optical system for the detection of
gravitational waves in space are computationally expensive. By comparison,
Eqs. 10a, 10b, 11c and 11f are computationally simple. The second advantage is
that the finite decomposition for MEM can eliminate the divergence problem of
MSD spot size for diffracted beams even though the MSD spot size is dependent
on the decomposition parameters. The paper by S. N. Vlasov et al. [11] shows
the MSD spot size for the non-diffracted beam is a half-hyperbola function of
the propagation distance. The MSD spot size for MEM beams here can be regarded
as changing the intensity profile rather than changing the integral limitation
of Eq. 2. Based on this fact, we can use the same method used by S. N. Vlasov
et al. [11] and M. A. Porras et al. [4] to prove the MSD spot size and energy
center for the MEM beam here still is the half-hyperbola function of the
propagation distance and obeys the ABCD law in the ABCD system in diffraction
cases. This is the third advantage.

##  3 Validation of MSD spot size definition and numerical code

In this section, we will subject the MSD spot size definition to tests in
cases of a simple astigmatic Gaussian beam as well as Nearly-Gaussian beams
which can appear in laboratory experiments. At the same time, the tests serve
to calibrate our numerical code. The numerical calculation of analytical
expression in this paper is performed by C++ code with double type precision.

###  3.1 Simple astigmatic Gaussian beam

The MSD spot size definition agrees with the definition of the spot size for
Gaussian, HG and LG beams [14, 15]. If a Gaussian beam impacts on a curved
surface at an angle, the beam becomes a simple astigmatic Gaussian beam, which
has an elliptical spot [19, 20]. Considering the simple astigmatic Gaussian
beam as an expansion of the basic Gaussian and HG beams, the MSD spot size
should agree with the analytic spot size here.

The electric field of the input simple astigmatic Gaussian beam is [21]

$\begin{split}E_{in}=u_{in}\exp{\left(i\omega
t\right)}=&\sqrt{\frac{P}{\pi}}\frac{1}{\sqrt{w_{x}w_{y}}}\exp{(-\frac{x^{2}}{w_{x}^{2}}-\frac{y^{2}}{w_{y}^{2}})}\\\
&\times\exp{\left(i\left(-k\left(\frac{x^{2}}{2R_{x}}+\frac{y^{2}}{2R_{y}}\right)+\frac{1}{2}\left(\zeta_{x}+\zeta_{y}\right)-kz+\omega
t\right)\right)}.\end{split}$ (13)

Here, the beam propagates along $+z$ direction and the parameter $w_{x,y}$ is
the spot size for x or y direction of the simple astigmatic Gaussian beam. It
is defined as
$w_{x,y}=w_{0_{x,y}}\sqrt{1+\left(\frac{z}{z_{r_{x,y}}}\right)^{2}}$, using
the Rayleigh range $z_{r_{x,y}}=\frac{\pi w_{0_{x,y}}^{2}}{\lambda}$ and the
waist $w_{0_{x,y}}$, where $\lambda$ is the wavelength and
$k=\frac{2\pi}{\lambda}$ is the wave number. The radius of curvature is
$R_{x,y}=z\left(1+\left(\frac{z_{r_{x,y}}}{z}\right)^{2}\right)$ and the Gouy
phase is $\zeta_{x,y}=\arctan\frac{z}{z_{r_{x,y}}}$. The total power of this
beam is $P$. Here we set $w_{0_{x}}=$1\text{\,}\mathrm{m}\mathrm{m}$$,
$w_{0_{y}}=$3\text{\,}\mathrm{m}\mathrm{m}$$, $P=1$ and
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$. This beam can be decomposed
into the superposition of a finite number of HG modes by using Eq. 4. Here we
use two different waists $0.8\text{\,}\mathrm{m}\mathrm{m}$ and
$0.6\text{\,}\mathrm{m}\mathrm{m}$ for the basic HG modes. The direction and
center of these HG modes are the same as the input beam. The decomposition
happens in the waist plane.

Figure 1: (a). The intensity profile for
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$. (b). The estimated values of MSD spot
size for different max mode order $N$ compared with the analytic spot size for
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$. (c). The power ratio within the MSD
spot size for different max mode order $N$ for
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$. (d). The estimated values of spot
size of MEM beam compared with the analytic spot size for different
propagation distances.

Figure 1 shows the intensity profile for
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$ of the MEM beam which represents the
simple astigmatic Gaussian beam. The yellow line in Fig. 1 is the boundary of
the MSD spot for the MEM beam and the green point is the estimated beam center
$(0,0)$. The waist of the basic HG modes for the MEM beam is
$w_{0}=$0.6\text{\,}\mathrm{m}\mathrm{m}$$ and the max mode order is $N=200$.
Figures 1 and 1 show the variations for the MSD spot size and the power ratio
within the MSD spot range with different max mode order $N$. The waist of the
basic HG modes for the MEM beam is $w_{0}=$0.6\text{\,}\mathrm{m}\mathrm{m}$$
or $w_{0}=$0.8\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$. The blue points in Fig. 1 shows the
MSD spot size for different max mode order $N$ for
$w_{0}=$0.8\text{\,}\mathrm{m}\mathrm{m}$$ in x direction, the red points show
the MSD spot size for $w_{0}=$0.6\text{\,}\mathrm{m}\mathrm{m}$$ in x
direction, the green points show the MSD spot size for
$w_{0}=$0.8\text{\,}\mathrm{m}\mathrm{m}$$ in y direction and the purple
points show the MSD spot size for $w_{0}=$0.6\text{\,}\mathrm{m}\mathrm{m}$$
in y direction. The black line and orange line in Fig. 1 show the analytic
spot size for this simple astigmatic Gaussian beam in x direction and y
direction respectively. The purple points in Fig. 1 show the power ratio
within the MSD spot range for different max mode orders N for
$w_{0}=$0.8\text{\,}\mathrm{m}\mathrm{m}$$, the blue points show the power
ratio within the MSD spot range for $w_{0}=$0.6\text{\,}\mathrm{m}\mathrm{m}$$
and the orange line shows the power ratio inside the circular spot range for
the Gaussian beam. When the max mode order $N$ is large enough, the MSD spot
size is coincident with the analytical spot size of the input simple
astigmatic Gaussian beam, no matter what the waist of the basic HG modes is.
As shown in Fig. 1, the MSD spot size is coincident with the analytic spot
size for the simple astigmatic Gaussian beam for different $z$. The max mode
order $N$ is 200 and the waist of the basic HG modes $w_{0}$ is
$0.6\text{\,}\mathrm{m}\mathrm{m}$. The red line in Fig. 1 shows the MSD spot
size for different $z$ in x direction, the green line shows the MSD spot size
in y direction, the blue line shows the analytical spot size in x direction
and the orange line shows the analytical spot size in y direction. The MSD
spot size is a hyperbolic function of $z$. The fractional beam energy
concentrated inside the MSD spot range of this situation is $92.15\%$. If the
max mode order $N$ is large enough, the waist $w_{0}$ of the basic HG modes
will have little influence on the estimated value of MSD spot size and the
power ratio within the MSD spot range.

Another non-diffracted beam concerning a shifted HG beam will be discussed in
B.

###  3.2 Nearly-Gaussian beams

In this subsection, we will test the MSD spot size definition for some non-
perfect beams as they can appear in laboratory experiments. We use the MEM to
decompose these non-perfect beams and then calculate the MSD spot size and the
fractional beam energy concentrated inside the MSD spot range for these beams.

####  3.2.1 Halo Beam

A halo beam might be produced by touching a fiber end to a monolithic fiber
collimator with optical contact gel in the laboratory. If there is a small gap
between the collimator and the fiber, the intensity profiles will deform like
a halo as illustrated in Fig. 2:

Figure 2: (a) The intensity profile for
$z=$75.865\text{\,}\mathrm{m}\mathrm{m}$$ and the yellow line is the boundary
of the MSD spot. (b) The intensity profile for
$z=$-362.135\text{\,}\mathrm{m}\mathrm{m}$$ and the yellow line is the
boundary of the MSD spot.

The ”plus” points shown in Figs. 2 and 2 mark the energy centers. The energy
center in Fig. 2 is $(-0.04281,0.03558)$. It is $(-0.03492,0.02280)$ in Fig.
2. The fractional beam energy concentrated inside the spot size shown in Fig.
2 is $81.2142\%$ and $84.0180\%$ in Fig. 2. Figure 3 shows the MSD spot size
for different $z$ and Fig. 3 shows the fractional beam energy concentrated
inside the MSD spot for different $z$. The MSD spot size for the halo beam
satisfies the hyperbolic law and the fractional beam energy concentrated
inside the MSD spot size is always bigger than $80.94\%$. The waist of these
HG modes is $w_{0}=$644.314\text{\,}\mu\mathrm{m}$$ and the waist plane of
these HG modes are all located in the cross-section of
$z=$0\text{\,}\mathrm{m}\mathrm{m}$$. The decomposition results of the MEM
beam for this halo beam are shown in D.

Figure 3: (a) The MSD spot size of halo beam for different $z$. (b)The
fractional beam energy concentrated inside the MSD spot for different $z$.

####  3.2.2 Nearly-Gaussian beam II

Another non-perfect beam is shown in Fig. 4.

Figure 4: (a) The intensity profile for
$z=$481.101\text{\,}\mathrm{m}\mathrm{m}$$ and the yellow line is the boundary
of the MSD spot. (b) The intensity profile for
$z=$1081.101\text{\,}\mathrm{m}\mathrm{m}$$ and the yellow line is the
boundary of the MSD spot.

The ”plus” points shown in Figs. 4 and 4 mark the energy centers. The energy
center in Fig. 4 is $(-0.0691484,-0.0196754)$. It is $(-0.0927757,-0.0306717)$
in Fig. 4. The fractional beam energy concentrated inside the spot size shown
in Fig. 4 is $89.6420\%$ and $88.0395\%$ in Fig. 4. Figure 5 shows the MSD
spot size for different $z$ and Fig. 5 shows the fractional beam energy
concentrated inside the MSD spot for different $z$. The waist of these HG
modes is $w_{0}=$399.465\text{\,}\mu\mathrm{m}$$ and the waist planes of these
HG modes are all located in the cross-section of
$z=$0\text{\,}\mathrm{m}\mathrm{m}$$. The decomposition results of the MEM
beam for this beam are shown in E.

Figure 5: (a) The MSD spot size of nearly-Gaussian beam II for different $z$.
(b)The fractional beam energy concentrated inside the MSD spot for different
$z$.

##  4 MSD spot size and power ratio within the MSD spot range for top-hat
beams in detection of gravitational waves in space.

In this section, we shall enter the core of our work and trace the evolution
of the spot size of a top-hat beam propagating within an optical bench for a
LISA or TAIJI type mission. The spot size together with the corresponding
power ratio within this spot range will also be estimated. To this end, we
shall first address in the following subsection the issue of hard edge
diffraction of a flat top beam. The MSD spot size for a top-hat beam in a
science interferometer in a LISA or TAIJI type optical bench will then be
simulated. Some error analysis will then be presented to render our
calculations more trustworthy.

###  4.1 Spot size for a top-hat beam

In this section, we investigate the performance of the MSD spot size in the
top-hat beam case. Unlike the previously discussed cases, diffraction occurs
at the edge of the beam and this generates divergence for the MSD spot size of
a top-hat beam. The analysis will then be applied in the next subsection when
we look at the propagation of a flat top beam within a distance scale dictated
by the size of an optical bench in detection of gravitational waves in space.

Consider a top-hat beam which may be regarded as a plane wave truncated by a
circular aperture. The complex amplitude of the top-hat beam maybe written as

$u_{top}(x,y,0)=\left\\{\begin{aligned} &\sqrt{\frac{P}{\pi}}\frac{1}{r},\
x^{2}+y^{2}\leq r^{2}\\\ &0,\ x^{2}+y^{2}>r^{2},\end{aligned}\right.$ (14)

where $r$ is the initial radius of the top-hat beam and $P$ is the total
power. Here we assume the beam propagates along $+z$ direction and the
wavelength of this top-hat beam is
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$.

As discussed in Section 2, the top-hat beam can be decomposed by MEM and the
propagation of this MEM beam can be used to represent the propagation of the
top-hat beam. The beam center and the propagation direction of these basic HG
modes are the same as the top-hat beam in this section. The waist planes of
these basic HG modes are located at the initial plane of the top-hat beam. The
total power of the input top-hat beam is $P=1$ a.u.. The initial radius $r$ of
the top-hat beam is $2.5\text{\,}\mathrm{m}\mathrm{m}$ here which is
coincident with the top-hat beam in the paper by L. D’Arcio et al.[1]

One obvious shortcoming of the MSD spot size for the hard-edge diffraction
beam is, that it is divergent [5]. The word divergent here means that the MSD
spot size for the hard-edge diffraction beam is infinite (Eq.
2$\rightarrow\infty$). Rounding off the spatial intensity distribution in some
reasonable way such as with a super-Gaussian edge is one choice to solve this
divergence [5]. Alternatively, another choice is to truncate the limitation of
the integration for MSD spot size to eliminate the large angle and evanescent
waves [5]. Many criteria are proposed for the truncated limitation such as
general truncated second-order moment method [6], asymptotic approximation
method [7] and self-convergent beam width method [8] and many more. However,
all of them introduce another free parameter which is changed by hand and can
influence the final value of the MSD spot size. There is no general guidance
to find a suitable value for the free parameter. These truncated methods
introduce another disadvantage that they can not guarantee that the hyperbolic
law and ABCD law for propagation still holds.

The MSD spot size is sensitive to the diffracted wings even if there are only
few fractional parts of beam energy inside these wings [7]. Limiting the
interval of integration of the MSD spot size definition to eliminate the
divergence [7] can be regarded as only reserving a few dominating wings. The
diffracted beam may also be decomposed by MEM as the previous section shows.
The higher the HG mode order is, the more outside wings can be represented. In
a sense, using a finite number of HG modes to represent the diffracted beam is
similar to limiting the interval of the integration. Compared with the MSD
spot size of the diffracted beam obtained by limiting the upper and lower
limits of the integral, the advantages of the MSD spot of the MEM beam are
that it still satisfies the ABCD law during passing through the ABCD system
and still exhibits the hyperbolic function of propagation distance.

Figures 6, 6 and 6 show the intensity profiles of the MEM beams which
represent the top-hat beam for the propagation distances
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ ,
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$. The waist of the basic HG modes of
the MEM beam is $w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$ and the max mode
order $N$ of the MEM beams is $N=200$. Figures 6, 6 and 6 show the intensity
distributions of the MEM beams along the x-axis of Figs. 6, 6 and 6. The
vertical lines in these subfigures are the edges of the MSD spot range for
different MEM beams. The blue vertical lines in these subfigures represent the
MSD spot size for the MEM beam with
$w_{0}=$0.25\text{\,}\mathrm{m}\mathrm{m}$$, the red lines represent
$w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$, the green lines represent
$w_{0}=$0.75\text{\,}\mathrm{m}\mathrm{m}$$, the purple lines represent
$w_{0}=$1.0\text{\,}\mathrm{m}\mathrm{m}$$ and the orange lines represent
$w_{0}=$1.25\text{\,}\mathrm{m}\mathrm{m}$$. The dark-red vertical line is
special and represents the spot size $w_{f}$ calculated from the paper by E.
M. Drège et al.[9] A typical divergence angle $\theta_{f}$ is defined as the
distance between the central brightest point and the point where the intensity
is $\frac{1}{e^{2}}$ of the maximum intensity value over the large propagation
distance $z$. Furthermore, the spot size $w_{f}$ is defined as the divergence
angle $\theta_{f}$ times the propagation distance $z$. This means the spot
size $w_{f}$ represents the distance between the central brightest point and
the point where the intensity is $\frac{1}{e^{2}}$ of the maximum intensity
value in the far-field. In the far-field, this spot size $w_{f}$ and the far-
field divergence $\theta_{f}$ angle is a function of wavelength $\lambda$ and
the radius of the circular aperture $R$ [9]

$\displaystyle\theta_{f}=\frac{2.5838\lambda}{2\pi R},$ (15a) $\displaystyle
w_{f}=\theta_{f}\cdot z.$ (15b)

The max mode orders $N$ in the above cases are fixed to be 200. As a matter of
fact, the max mode order $N$ may also influence the MSD spot size of a top-hat
beam. Figures 7, 7 and 7 show the variation of the spot size with the max mode
order $N$ for the propagation distances
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$, $z=$600\text{\,}\mathrm{m}\mathrm{m}$$
and $z=$3000\text{\,}\mathrm{m}\mathrm{m}$$. The variation of the
corresponding power ratio within the MSD spot range are shown in Figs. 7, 7
and 7 for the propagation distances $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ ,
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$.

As mentioned before, the MSD spot size is infinite in hard-edge diffraction
situations such as the top-hat beam here. Using the MEM, we can get a finite
MSD spot size for the diffracted beam. However, this finite MSD spot size is
dependent on the max mode order $N$ and the waist $w_{0}$ of the basic HG
modes. For any fixed propagation distance $z$, the MSD spot size diverges with
the increasing mode order $N$. As Fig. 6 shows, all these spot ranges are
large enough to contain the main part of the intensity patterns except for
$w_{f}$. The spot size calculated from the analytical divergence angle of the
top-hat beam $w_{f}$ is so small that we can not use it here.

Figure 6: The intensity profiles of the simulated top-hat beam for the
propagation distance $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (a),
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ (c) or
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$ (e). The intensity distributions of
the simulated top-hat beam along x-axis for the propagation distance
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (b),
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ (d) or
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$ (f).

Figure 7: The MSD spot size of the simulated top-hat beam for different max
mode order $N$ with the propagation distance
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (a),
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ (c) or
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$ (e). The power ratio within the MSD
spot range of the simulated top-hat beam for different max mode order $N$ with
the propagation distance $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (b),
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$ (d) or
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$ (f).

The Fresnel number for this top-hat beam is

$F(z)=\frac{r^{2}}{z\lambda}.$ (16)

The Fresnel numbers for propagation distances
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$, $z=$600\text{\,}\mathrm{m}\mathrm{m}$$
and $z=$3000\text{\,}\mathrm{m}\mathrm{m}$$ are $F(100)=58.7406$,
$F(600)=9.7901$, $F(3000)=1.9580$. All of these three cross-sections are
located in the near field of this top-hat beam because the Fresnel numbers are
bigger than $1$. When the max mode order $N$ is large enough, the estimated
spot ranges for these MEM beams are almost the same for very near fields such
as $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$600\text{\,}\mathrm{m}\mathrm{m}$$. As for the not very near field such as
$z=$3000\text{\,}\mathrm{m}\mathrm{m}$$, the estimated spot ranges for
different MEM beams are obviously different. The differences of the power
ratio within the corresponding MSD spot range are small for
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$, $z=$600\text{\,}\mathrm{m}\mathrm{m}$$
and $z=$3000\text{\,}\mathrm{m}\mathrm{m}$$. The larger the waist for the
basic HG modes is, the smaller the MSD spot size and power ratio within the
MSD spot range are. Diffracted aberration Gaussian beam as another diffracted
beam case will be discussed in C.

###  4.2 Spot size estimation for a top-hat beam in Science Interferometer

In this subsection, we are in a position to numerically track the evolution of
the spot size of a flat top beam for the science interferometer in an optical
bench for a LISA or TAIJI type mission. Further we will also argue that,
despite of the divergence behaviour of the MSD spot size for a flat top beam,
our analysis still draws a reliable conclusion.

Consider one of the optical designs for the LISA optical bench [1] and we take
a top-hat beam of a science interferometer as a representative example of
optical beam propagation within the optical bench for our investigation. In
Fig. 8, the science interferometer subsystem of this optical bench is shown.
The initial radius of the input top-hat beam in the exit pupil plane is
$2.5\text{\,}\mathrm{m}\mathrm{m}$ [1]. Given that the optical bench design of
neither of both missions is available yet, we can only estimate that the
propagation distances in the science interferometer on the optical bench will
be in the order of tens of centimeters. Consequently, we test here the spot
size evolution for propagation distances up to
$60\text{\,}\mathrm{c}\mathrm{m}$. We use Eqs. 11 and 12 to calculate the MSD
spot size and the corresponding power ratio within the spot range here.

Figure 8: Science interferometer qualitatively derived from paper[1], shown
only starting from the $45$ degree component delivering the received beam from
the telescope onto the optical bench, neglecting imaging systems and
redundancy.

Figures 9, 9 and 9 show the MSD spot sizes for different propagation distances
$z$ of different MEM beams which represent the top-hat beam with initial radii
$r=$2\text{\,}\mathrm{m}\mathrm{m}$$, $2.5\text{\,}\mathrm{m}\mathrm{m}$ or
$3\text{\,}\mathrm{m}\mathrm{m}$ respectively. The settings of these MEM beams
are the same except for the waist size $w_{0}$ of basic HG modes in the same
subfigure. The max mode order $N$ of these MEM beams is equal to $200$. The
origin and direction of the basic HG modes of MEM beams are the same as the
origin and direction of the input top-hat beam.

Figures 9, 9 and 9 show the power ratio within the MSD spot range for
different propagation distances $z$ of different MEM beams which represent the
top-hat beam with initial radii $r=$2\text{\,}\mathrm{m}\mathrm{m}$$,
$2.5\text{\,}\mathrm{m}\mathrm{m}$ or $3\text{\,}\mathrm{m}\mathrm{m}$
respectively. The settings of these MEM beams are the same as the settings in
Figs. 9, 9 and 9.

For certain MEM beams, the MSD spot size increases with the propagation
distance and it’s the hyperbolic function of propagation distance $z$ [17].
For certain propagation distances $z$, the MSD spot size increases as the
waist size $w_{0}$ of the basic HG modes of the MEM beam decreases. As we
discussed before, even though the MSD spot size of the MEM beam representing
diffracted beams is finite, it is not unique for the corresponding diffracted
beam. The MSD spot size growth rate increases with propagation distance.
Furthermore, this growth rate increases as the waist size $w_{0}$ of basic HG
beams decreases or the initial radius $r$ of the top-hat beam decreases.

For certain MEM beams, the power ratio within the MSD spot range decreases as
the propagation distance $z$ increases here. For certain propagation distances
$z$, the power ratio within the MSD spot range decreases as the waist size
$w_{0}$ of basic HG modes increases. The growth rate of the power ratio within
the MSD spot range increases with decreasing initial radii $r$ of the top-hat
beam.

Figure 9: The MSD spot sizes for different propagation distance $z$ for the
top-hat beams with initial radii $r=$2.0\text{\,}\mathrm{m}\mathrm{m}$$
(a),$r=$2.5\text{\,}\mathrm{m}\mathrm{m}$$ (c) or
$r=$3.0\text{\,}\mathrm{m}\mathrm{m}$$ (e). The power ratio within the MSD
spot range for different propagation distance $z$ of the top-hat beams with
different initial radii $r=$2.0\text{\,}\mathrm{m}\mathrm{m}$$
(b),$r=$2.5\text{\,}\mathrm{m}\mathrm{m}$$ (d) or
$r=$3.0\text{\,}\mathrm{m}\mathrm{m}$$ (f).

###  4.3 Error analysis

In the previous section, we qualitatively discussed the performance of MSD
spots for a science interferometer. In this part, we use error analysis to
quantitatively discuss the performance of MSD spot size. The estimated errors
of the MSD spot size and the estimated errors of the power ratio within the
MSD spot range are small enough to be negligible for the top-hat beam initial
radius from $2$-3\text{\,}\mathrm{m}\mathrm{m}$$ in the propagation distance
range $z\in[0,600]$ $\text{\,}\mathrm{m}\mathrm{m}$. The power ratios within
the MSD spot ranges are large enough for the top-hat beam initial radius from
$2$-3\text{\,}\mathrm{m}\mathrm{m}$$ in the propagation distance range
$z\in[0,600]$ $\text{\,}\mathrm{m}\mathrm{m}$. This means that the estimated
values of MSD spot size work well in a science interferometer in the detection
of gravitational waves in space.

Unphysical decomposition parameters such as the waist $w_{0}$ of basic HG
modes can influence the value of the MSD spot size of an MEM beam. It is
necessary to estimate the effect of the unphysical decomposition parameter
$w_{0}$ on the MSD spot size of a top-hat beam in a science interferometer.
Here we define the estimated error of the MSD spot size of the top-hat beam as
twice the ratio of the difference between the maximum value of the MSD spot
size for a certain propagation distance and the minimum value of the MSD spot
size for the same propagation distance to the sum of this maximum value and
minimum value

$err_{spot}(z)=2\frac{w_{max}(z)-w_{min}(z)}{w_{max}(z)+w_{min}(z)},$ (17)

where $w_{max}(z)$ is the maximum value of the MSD spot size for propagation
distance $z$ and $w_{min}(z)$ is the minimum value of the MSD spot size for
propagation distance $z$.

Similar to the estimated error definition of the MSD spot, we define the
estimated error of the power ratio within the MSD spot range as

$err_{p_{r}}(z)=2\frac{p_{r_{max}}(z)-p_{r_{min}}(z)}{p_{r_{max}}(z)+p_{r_{min}}(z)},$
(18)

where $p_{r_{max}}(z)$ is the maximum value of the power ratio within the MSD
spot range for propagation distance $z$ and $p_{r_{min}}(z)$ is the minimum
value of the power ratio within the MSD spot range for propagation distance
$z$.

Figure 10: (a) The estimated error of MSD spot sizes for different propagation
distance $z$. (b) The estimated error of the power ratio within the MSD spot
range for different propagation distance $z$.

Figure 10 shows the estimated error of the MSD spot size for different
propagation distances $z$ of different MEM beams which represent the top-hat
beam with the initial radii $r=$2\text{\,}\mathrm{m}\mathrm{m}$$,
$2.5\text{\,}\mathrm{m}\mathrm{m}$ and $3\text{\,}\mathrm{m}\mathrm{m}$. It is
obvious that the estimated error of MSD spot size for the top-hat beam
increases with the propagation distance $z$. The maximum estimated error of
MSD spot size in these simulations of the top-hat beam with
$2\text{\,}\mathrm{m}\mathrm{m}$ initial radius is $5.764\%$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$ and the max value of MSD spot size in
Fig. 9 is $2.157\text{\,}\mathrm{m}\mathrm{m}$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$. For the top-hat beam with
$2.5\text{\,}\mathrm{m}\mathrm{m}$ initial radius, the maximum estimated error
of MSD spot size is $2.705\%$ for $z=$60\text{\,}\mathrm{c}\mathrm{m}$$ and
the max value of MSD spot size in Fig. 9 is
$2.58\text{\,}\mathrm{m}\mathrm{m}$ for $z=$60\text{\,}\mathrm{c}\mathrm{m}$$.
As for the top-hat beam with $3\text{\,}\mathrm{m}\mathrm{m}$ initial radius,
the maximum estimated error of MSD spot size is $1.514\%$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$ and the max value of MSD spot size in
Fig. 9 is $3.044\text{\,}\mathrm{m}\mathrm{m}$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$. The max estimated error of the MSD spot
size decreases when the initial radius of the top-hat beam increases. For the
science interferometer that we investigated here, the initial radius $r$ of
the input top-hat beam is $2.5\text{\,}\mathrm{m}\mathrm{m}$, and the
corresponding maximum estimated error of MSD spot size is $2.705\%$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$. The estimated error of the MSD spot
size for this situation is so small that we can use this MSD spot size to
represent the spot size of the top-hat beam here. If the initial radius r is
changed to $2\text{\,}\mathrm{m}\mathrm{m}$ or
$3\text{\,}\mathrm{m}\mathrm{m}$, the estimated error of the MSD spot size
would still be small enough to use.

Figure 10 shows the estimated errors of the power ratio within the MSD spot
range for different propagation distances $z$ of different MEM beams which
represent the top-hat beams with the initial radii
$r=$2\text{\,}\mathrm{m}\mathrm{m}$$, $2.5\text{\,}\mathrm{m}\mathrm{m}$ and
$3\text{\,}\mathrm{m}\mathrm{m}$. The maximum estimated error of the power
ratio within the MSD spot range of the top-hat beam with
$2\text{\,}\mathrm{m}\mathrm{m}$ initial radius is $1.481\%$ for
$z=$60\text{\,}\mathrm{c}\mathrm{m}$$. For the top-hat beam with
$2.5\text{\,}\mathrm{m}\mathrm{m}$ initial radius, the maximum estimated error
of the power ratio within the MSD spot range of the top-hat beam is $0.959\%$
for $z=$60\text{\,}\mathrm{c}\mathrm{m}$$. As for the top-hat beam with
$3\text{\,}\mathrm{m}\mathrm{m}$ initial radius, the maximum estimated error
of MSD spot size is $0.951\%$ for $z=$0\text{\,}\mathrm{c}\mathrm{m}$$. The
fractional part of the beam energy concentrated inside the MSD spot range is
always larger than $94.326\%$ for the top-hat beam with
$2\text{\,}\mathrm{m}\mathrm{m}$ initial radius in Fig. 9, $95.351\%$ for the
top-hat beam with $2.5\text{\,}\mathrm{m}\mathrm{m}$ initial radius in Fig. 9
and $95.66\%$ for the top-hat beam with $3\text{\,}\mathrm{m}\mathrm{m}$
initial radius in Fig. 9. The maximum estimated error of the power ratio
within the MSD spot range for the top-hat beam with
$2.5\text{\,}\mathrm{m}\mathrm{m}$ initial radius is the smallest in these
three cases. If the initial radius $r$ is changed from
$2.5\text{\,}\mathrm{m}\mathrm{m}$ to $2\text{\,}\mathrm{m}\mathrm{m}$ or
$3\text{\,}\mathrm{m}\mathrm{m}$, the estimated error of the MSD spot size
would still be small enough to use.

There are two reasons why the power inside the MSD spot range does not equal
the power of the input top-hat beam. First of all, the power loss appears in
the MEM process because of the finite decomposition mode order. This power
loss is independent of the propagation distance $z$ and it increases with the
waist size $w_{0}$ of basic HG modes. Secondly, a top-hat beam is a diffracted
beam, parts of the diffracted wings always appear outside a finite range. This
power loss is dependent on the propagation distance $z$. At the same time, we
notice that the MSD spot size increases with the propagation distance $z$. As
mentioned before, the power ratio within the MSD spot range decreases as the
propagation distance $z$ increases. This means that the rate of increase of
the MSD spot size is slightly slower than the diffusion rate of the diffracted
wings.

In the above discussion, we find that the estimated errors of the MSD spot
size, as well as the estimated errors of the power ratio within the MSD spot
range, are small and the power ratios within the MSD spot range are large
enough for the top-hat beam initial radius from
$2$-3\text{\,}\mathrm{m}\mathrm{m}$$ in the propagation distance range
$z\in[0,600]$ $\text{\,}\mathrm{m}\mathrm{m}$.

The optical elements on the LISA Pathfinder optical bench had a size of
$15\times 20\times 7$\text{\,}\mathrm{m}\mathrm{m}$$ [22]. The largest spot
size for the top-hat beam in a science interferometer here is
$2.580\text{\,}\mathrm{m}\mathrm{m}$. The usual practice of the optical
element’s half width is designed to be approximately equal or slightly larger
than three times the spot sizes of the input beams to avoid the clipping
problem [5]. Using this rule, the size of the optical elements can be
$15.48\text{\,}\mathrm{m}\mathrm{m}$. Based on this size and the size of the
optical elements for LISA Pathfinder, we suggest that also space-based
gravitational wave detectors like LISA and TAIJI will use comparable component
sizes.

Depending on the optical designs, i.e. particularly the aperture sizes and
potential use of imaging optics, the given equations in Section 2 can be used
to derive the optimal size of all mirrors and beam splitters. This optimal
size means that the components are small and lightweight but at the same time,
large enough to prevent beam clipping. The present work will serve as a useful
guide in the future system design of the optical bench and the sizes of the
optical components.

##  5  Conclusion

In this paper, a slightly different variant of the definition of spot size is
proposed, with which we calculate the MSD spot size of a light beam for
arbitrary propagation distances. Even with diffraction taken into account, the
definition still upholds the hyperbolic law and ABCD law of the MSD spot size.
Though the MSD spot size for a diffracted MEM beam is dependent on the
decomposition parameters such as the max mode order and the waist of the basic
HG modes, it is shown that, for different reasonable choices of parameters,
the estimated error for the MSD spot size and the power ratio within the MSD
spot range will have little effect on our drawn conclusion. With the help of
this method, optical systems such as the interferometers in space based
gravitational wave detectors can be efficiently tested for beam clipping.

Our study results provide a method to allow for an optimal design of optical
systems with top-hat or other types of non-Gaussian beams. Furthermore, it
allows testing the interferometry of space-based gravitational wave detectors
for beam clipping in optical simulations. The present work will serve as a
useful guide in the future system design of the optical bench and the sizes of
the optical components.

##  6 Acknowledgements

We thank Prof. Yun-Kau Lau and Apl. Prof. Gerhard Heinzel for the useful
discussions. This work has been supported in part by the National Key Research
and Development Program of China under Grant No.2020YFC2201501, the National
Science Foundation of China (NSFC) under Grants No. 12147103 (special fund to
the center for quanta-to-cosmos theoretical physics), No. 11821505, the
Strategic Priority Research Program of the Chinese Academy of Sciences under
Grant No. XDB23030100, and the Chinese Academy of Sciences (CAS) and the Max
Planck Society (MPG) in the framework of the LEGACY cooperation on low-
frequency gravitational wave astronomy (M.IF.A.QOP18098). Likewise, we
gratefully acknowledge the German Space Agency, DLR and support by the Federal
Ministry for Economic Affairs and Energy based on a resolution of the German
Bundestag (FKZ50OQ1801) as well as the Deutsche Forschungsgemeinschaft (DFG)
funding the Cluster QuantumFrontiers (EXC2123, Project ID 390837967) for
funding the work contributions by Gudrun Wanner. We gratefully acknowledge DFG
for funding the Collaborative Research Centres CRC 1464: TerraQ – Relativistic
and Quantum-based Geodesy and the Clusters of Excellence PhoenixD (EXC 2122,
Project ID 390833453). In this paper, the part of the numerical computation is
finished by TAIJI Cluster.

##  Appendix A power ratio within the MSD spot range

Here, we deduce the relationship between the power ratio within the MSD spot
range and $z$. The intensity of the MEM beam can be represented as

$\begin{split}I=&\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}|a_{m_{1}n_{1}}|^{2}|u_{m_{1}n_{1}}(x,y,z)|^{2}+2\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}\sum\limits_{m_{2}=0}^{N}\sum\limits_{n_{2}=0}^{N-m_{2}}|a_{m_{1}n_{1}}||a_{m_{2}n_{2}}|\\\
&\cdot|u_{m_{1}n_{1}}(x,y,z)||u_{m_{2}n_{2}}(x,y,z)|\\\
&\cdot\cos{(\beta_{m_{1}n_{1}}+\phi_{m_{1}n_{1}}-\beta_{m_{2}n_{2}}-\phi_{m_{2}n_{2}})}\quad|\text{for}\
(m_{2}\neq m_{1}\ \text{or}\ n_{2}\neq n_{1}),\end{split}$ (19)

where $(m_{1},n_{1})$, $(m_{2},n_{2})$ are two of the basic HG modes of the
MEM beam and $N$ is the max mode order of the MEM beam. The integrand of Eq.
12 is

$\begin{split}\frac{I}{P_{MEM}}=&\frac{1}{\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}|a_{m_{1}n_{1}}|^{2}}\frac{1}{w(z)^{2}}\exp{\left(-2\frac{x^{2}+y^{2}}{w(z)^{2}}\right)}\\\
&\cdot\left(\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}|a_{m_{1}n_{1}}|^{2}c_{m_{1}n_{1}}^{2}H_{m_{1}}^{2}\left(\frac{\sqrt{(}2)x}{w(z)}\right)H_{n_{1}}^{2}\left(\frac{\sqrt{(}2)y}{w(z)}\right)\right.\\\
&+2\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}\sum\limits_{m_{2}=0}^{N}\sum\limits_{n_{2}=0}^{N-m_{2}}|a_{m_{1}n_{1}}||a_{m_{2}n_{2}}|c_{m_{1}n_{1}}c_{m_{2}n_{2}}\\\
&\cdot
H_{m_{1}}\left(\frac{\sqrt{(}2)x}{w(z)}\right)H_{n_{1}}\left(\frac{\sqrt{(}2)y}{w(z)}\right)H_{m_{2}}\left(\frac{\sqrt{(}2)x}{w(z)}\right)H_{n_{2}}\left(\frac{\sqrt{(}2)y}{w(z)}\right)\\\
&\left.\cdot\cos{\left(\beta_{m_{1}n_{1}}-\beta_{m_{2}n_{2}}+\left(m_{1}+n_{1}-m_{2}-n_{2}\right)\zeta(z)\right)}\right).\end{split}$
(20)

Rewrite Eqs. 10a, 10b, 11c and 11f as

$\displaystyle s_{x}(z)=f_{1}(z)w(z),$ (21a) $\displaystyle
s_{y}(z)=f_{2}(z)w(z),$ (21b) $\displaystyle w_{x}(z)=f_{3}(z)w(z),$ (21c)
$\displaystyle w_{y}(z)=f_{4}(z)w(z).$ (21d)

$f_{1}(z)$, $f_{2}(z)$, $f_{3}(z)$ and $f_{4}(z)$ are dependent on $z$ because
of the Gouy phase part. Set $u=\frac{\sqrt{2}x}{w(z)}$ and
$v=\frac{\sqrt{2}y}{w(z)}$, then

$\begin{split}\frac{I}{P_{MEM}}=&\frac{I}{P_{MEM}}(u,v,z)=\frac{1}{\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}|a_{m_{1}n_{1}}|^{2}}\frac{1}{w(z)^{2}}\exp{\left(-u^{2}-v^{2}\right)}\\\
&\cdot\left(\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}|a_{m_{1}n_{1}}|^{2}c_{m_{1}n_{1}}^{2}H_{m_{1}}^{2}\left(u\right)H_{n_{1}}^{2}\left(v\right)\right.\\\
&+2\sum\limits_{m_{1}=0}^{N}\sum\limits_{n_{1}=0}^{N-m_{1}}\sum\limits_{m_{2}=0}^{N}\sum\limits_{n_{2}=0}^{N-m_{2}}|a_{m_{1}n_{1}}||a_{m_{2}n_{2}}|c_{m_{1}n_{1}}c_{m_{2}n_{2}}\\\
&\cdot
H_{m_{1}}\left(u\right)H_{n_{1}}\left(v\right)H_{m_{2}}\left(u\right)H_{n_{2}}\left(v\right)\\\
&\left.\cdot\cos{\left(\beta_{m_{1}n_{1}}-\beta_{m_{2}n_{2}}+\left(m_{1}+n_{1}-m_{2}-n_{2}\right)\zeta(z)\right)}\right).\end{split}$
(22)
$\displaystyle\int_{s_{x}(z)-w_{x}(z)}^{s_{x}(z)+w_{x}(z)}dx=\int_{\sqrt{2}(f_{1}(z)-f_{3}(z))}^{\sqrt{2}(f_{1}(z)+f_{3}(z))}\frac{w(z)}{\sqrt{2}}du,$
(23a)
$\displaystyle\int_{s_{y}(z)-w_{y}(z)}^{s_{y}(z)+w_{y}(z)}dy=\int_{\sqrt{2}(f_{2}(z)-f_{4}(z))}^{\sqrt{2}(f_{2}(z)+f_{4}(z))}\frac{w(z)}{\sqrt{2}}dv$
(23b)

Using Eq. 22 and Eq. 23 in the definition of the power ratio within the MSD
spot range Eq. 12, we find $\epsilon_{P_{spot}}$ is obviously dependent on $z$

$\epsilon_{P_{spot}}=\int_{\sqrt{2}(f_{1}(z)-f_{3}(z))}^{\sqrt{2}(f_{1}(z)+f_{3}(z))}\int_{\sqrt{2}(f_{2}(z)-f_{4}(z))}^{\sqrt{2}(f_{2}(z)+f_{4}(z))}\frac{w(z)^{2}}{2}\frac{I}{P_{MEM}}(u,v,z)dudv.$
(24)

The integrand $\frac{w(z)^{2}}{2}\frac{I}{P_{MEM}}(u,v,z)$ is still dependent
on $z$ because of the Gouy phase part. According to the property of the
$Arctan$ function, the Gouy phase always can be regarded as a constant when
$z$ is large enough. If $z$ is large enough, $f_{1}(z)$, $f_{2}(z)$,
$f_{3}(z)$ and $f_{4}(z)$, the integrand
$\frac{w(z)^{2}}{2}\frac{I}{P_{MEM}}(u,v,z)$ and $\epsilon_{P_{spot}}$ will be
independent from $z$.

##  Appendix B shifted HG beam

Sometimes, it’s not easy to find the center of the beam in the experiment. The
paper by X. Xue et al. [23] shows that people can use the intensity profiles
to find the superposition of a finite number of HG modes which can represent
the experimental beam. The centers of these HG modes are usually not the same
as the center of the input experimental beam. If the center of these
decomposed HG modes is set to the same as the origin of the experiment of the
beam, we would do a more accurate component analysis. Here, we would like to
test Eqs. 10a, 10b, 11c and 11f, especially the estimated value of beam center
in the shifted HG Gaussian beam situation. The shifted beam means that the
center of the beam is not the same as the centers of decomposed HG modes. For
instance, we set the order of this shifted HG beam as $(1,0)$, and its origin
is not located at the brightest point. Assume the waist of this shifted HG
beam is $w_{0}=$1\text{\,}\mathrm{m}\mathrm{m}$$, the propagation direction is
$(0,0,1)$ and the center is $(1,0)$. The electric field of this input shifted
HG beam is

$\begin{split}E_{in}=u_{in}\exp{\left(i\omega
t\right)}=&\sqrt{\frac{P}{\pi}}\frac{1}{w(z)}H_{1}\left(\frac{\sqrt{2}(x-1)}{w(z)}\right)H_{0}\left(\frac{\sqrt{2}y}{w(z)}\right)\exp{\left(-\frac{(x-1)^{2}+y^{2}}{w^{2}(z)}\right)}\\\
&\times\exp{\left(-ikz-ik\frac{(x-1)^{2}+y^{2}}{2R(z)}+i2\zeta(z)+i\omega
t\right)}.\end{split}$ (25)

These symbols are defined in Section 2. Here we use two different waists
$0.4\text{\,}\mathrm{m}\mathrm{m}$ and $0.5\text{\,}\mathrm{m}\mathrm{m}$ for
the finite number of HG modes. The direction of these HG modes is the same as
the input beam. The centers of these HG modes are $(0,0)$. The decomposition
happens in the waist plane of the shifted HG beam.

Figure 11: (a). The intensity profile for
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$. (b) .The intensity profile for
$z=$30\,000\text{\,}\mathrm{m}\mathrm{m}$$.

Figure 12: (a) The estimated values of spot size for different max decomposed
mode order superpositions of $z=$100\text{\,}\mathrm{m}\mathrm{m}$$. (b) The
estimated values of beam center for different max decomposed mode order
superpositions of $z=$100\text{\,}\mathrm{m}\mathrm{m}$$.(c). The power ratio
within the MSD spot size for different max mode order $N$ with
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$. (d). The estimated values of spot size
of MEM beam compared with the analytic spot size for different propagation
distances.

Figures 11 and 11 show the intensity profile for
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$30\,000\text{\,}\mathrm{m}\mathrm{m}$$ of the MEM beam which represent for
the shifted $(1,0)$ HG beam. In Fig. 11, the yellow line is the boundary of
the MSD spot for the MEM beam, the green point is the estimated beam center
$(1,0)$ and the red line is the coordinate axis. The waist of the basic HG
modes for the MEM beam is $w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$ and the
max mode order is $N=200$. The variations for the MSD spot size, energy center
and the power ratio within the MSD spot range with different max mode order
$N$ are shown in Figs. 12, 12 and 12. The waist of the basic HG modes for the
MEM beam is $w_{0}=$0.4\text{\,}\mathrm{m}\mathrm{m}$$ or
$w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$. As for Fig. 12, it clearly shows that
the MSD spot size is coincide with the analytic result for a HG beam with
different $z$.

The MSD spot size also is the hyperbolic function of $z$. The fractional beam
energy concentrated inside the MSD spot range of this situation is $94.77\%$.
If the max mode order $N$ is large enough, the waist $w_{0}$ of the basic HG
modes won’t influence the estimated value of MSD spot size and the power ratio
within the MSD spot range and the estimated value of MSD spot size and the
power ratio within the MSD spot range coincident with the analytic result.

##  Appendix C  diffracted aberration Gaussian beam

Aberration and diffraction are common phenomena that occur in imaging systems.
It is important to investigate the effects of the diffraction of optical
aberrations of Gaussian beams. In this section, we calculate the MSD spot size
for diffracted beams which are generated by passing aberrated Gaussian beams
through an aperture. We then assume a non-perfect Gaussian beam with
aberration $\phi_{R}$, which we model by using Zernike polynomials

$\displaystyle E_{aberr}(x,y,z)$
$\displaystyle=u_{00}(x,y,z)\exp{\left(iwt\right)}\exp{\left(ik\phi_{R}\right)}$
(26) $\displaystyle\phi_{R}$ $\displaystyle=\sum_{m}\sum_{n}Z_{n}^{m}$

Where $Z_{n}^{m}$ is the Zernike polynomial. A circular aperture is put in the
waist plane of the aberrated Gaussian beam and the beam passes through it
vertically. The aperture center is located on the Gaussian beam axis of this
aberrated beam and the coordinate value for aperture center is $(0,0,0)$. The
diffracted beam on the aperture plane is

$E_{diff}(x,y,0)=\left\\{\begin{aligned} &E_{aberr}(x,y,0),\ x^{2}+y^{2}\leq
r^{2}\\\ &0,\ x^{2}+y^{2}>r^{2}\end{aligned}\right..$ (27)

For simplicity, here we set the waist for $u_{00}(x,y,z)$ is
$w_{0}=$1\text{\,}\mathrm{m}\mathrm{m}$$,
$\lambda=$1064\text{\,}\mathrm{n}\mathrm{m}$$, the aperture radius
$r=w_{0}=$1\text{\,}\mathrm{m}\mathrm{m}$$ and the aberration phase is
$\phi_{R}=Z_{3}^{-3}=\sqrt{8}\rho^{3}\sin 3\phi$, where
$\rho=\sqrt{x^{2}+y^{2}}$ and $\phi$ is the corresponding azimuthal angle. We
decomposed the aberrated beam by Eq. 4 in the aperture plane and the
propagation of the diffracted beam can be represented by the propagation of
the decomposed MEM beam. We use five different waist settings for MEM to
produce five MEM decomposed beams. The settings are
$w_{0}=$0.1\text{\,}\mathrm{m}\mathrm{m}$$,
$w_{0}=$0.2\text{\,}\mathrm{m}\mathrm{m}$$,
$w_{0}=$0.3\text{\,}\mathrm{m}\mathrm{m}$$,
$w_{0}=$0.4\text{\,}\mathrm{m}\mathrm{m}$$ and
$w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$. All of these MEM beams have the
same max mode order $N=200$. The beam centers and the propagation directions
of the basic HG modes for MEM beams are all the same as the beam center and
the propagation direction of the diffracted aberration beam.

Figures 13 and 13 show the intensity profiles of the MEM beams which represent
the aberration beam for $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$. The waist of the basic HG modes of
the MEM beams is $w_{0}=$0.3\text{\,}\mathrm{m}\mathrm{m}$$ and the max mode
order $N$ of the MEM beams is $200$. Figures 13 and 13 show the intensity
distributions of the MEM beams along the y-axis of Figs. 13 and 13. The purple
vertical lines in these subfigures represent the MSD spot size for the MEM
beam with $w_{0}=$0.1\text{\,}\mathrm{m}\mathrm{m}$$, the red lines represent
$w_{0}=$0.2\text{\,}\mathrm{m}\mathrm{m}$$, the blue lines represent
$w_{0}=$0.3\text{\,}\mathrm{m}\mathrm{m}$$, the brown lines represent
$w_{0}=$0.4\text{\,}\mathrm{m}\mathrm{m}$$ and the orange lines represent
$w_{0}=$0.5\text{\,}\mathrm{m}\mathrm{m}$$. The max mode orders $N$ in the
above cases are fixed to 200. The variations of the MSD spot size with the max
mode order $N$ for $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$ are shown in Figs. 13 and 13
separately.

The maximum mode order $N$ of the MEM beam and the waist size $w_{0}$ of the
MEM basic HG modes affect the calculation of the spot size of the diffracted
aberration beam. As Fig. 13 shows, all these spot ranges are big enough to
contain the main part of the intensity patterns. The estimated spot ranges for
these MEM beams are almost the same for near propagation ranges such as
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$. As for
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$, the estimated spot ranges for
different MEM beams are obviously different. The larger the waist for the
basic HG modes is, the smaller the MSD spot size and power ratio within the
MSD spot range are. As mentioned before, the MSD spot size is divergent in
hard-edge diffraction situations such as the diffracted aberration beam here.
Using the MEM method, we can get a finite MSD spot size for diffracted
aberration beam. However, this finite MSD spot size is dependent on the waist
$w_{0}$ of the basic HG modes and the max mode order $N$ of the MEM beam.

Figure 13: The intensity profiles of the simulated gaussian beam with
aberration $\phi_{R}=Z_{3}^{-3}$ for the propagation distance
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (a) and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$ (d). The intensity distributions of
the aberration beam along y-axis for the propagation distance
$z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (b) and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$ (e). The MSD spot size of the
simulated aberration beam for different max mode order $N$ with the
propagation distance $z=$100\text{\,}\mathrm{m}\mathrm{m}$$ (c) and
$z=$1000\text{\,}\mathrm{m}\mathrm{m}$$ (f).

##  Appendix D The parameters of the MEM beam for the halo beam

The waist of the basic HG modes of the MEM beam for the halo beam is
$w_{0}=$644.314\text{\,}\mu\mathrm{m}$$. The corresponding complex
coefficients for different HG modes are shown in Appendix D. The HG modes in
Appendix D are ordered in the absolute amplitude of complex coefficient
$a_{mn}$ decreasingly.

complex coefficients for different HG modes of halo beam  mode HG(m,n) complex
coefficient $a_{mn}$ mode HG(m,n) complex coefficient $a_{mn}$ HG(0,0)
1.000000+0.000000i HG(1,2) -0.043850+0.030069i HG(0,2) -0.043689+0.208222i
HG(5,4) -0.013644+0.044648i HG(0,8) 0.197328+0.079289i HG(7,1)
0.041697+0.015981i HG(2,0) 0.059580+0.202141i HG(2,3) 0.034655-0.027614i
HG(8,0) 0.161853+0.090453i HG(4,5) 0.023136-0.037789i HG(6,0)
0.017052+0.166904i HG(3,6) -0.015738+0.037032i HG(7,0) -0.155352+0.053572i
HG(1,8) -0.003275+0.039806i HG(2,6) 0.138949+0.064269i HG(5,3)
0.033787+0.016435i HG(4,4) 0.131712+0.067135i HG(3,2) -0.035438-0.009784i
HG(0,6) -0.009895+0.144958i HG(4,1) 0.035344-0.005482i HG(0,4)
0.143830-0.009943i HG(3,5) 0.025634+0.022770i HG(6,2) 0.128264+0.065381i
HG(6,3) 0.021603-0.024983i HG(0,7) 0.121691-0.062886i HG(0,11)
-0.030008-0.013461i HG(4,2) -0.000903+0.128729i HG(1,5) -0.007792+0.030758i
HG(2,4) -0.007101+0.123730i HG(1,7) 0.021235+0.020630i HG(0,10)
0.092926-0.069031i HG(9,1) 0.024615-0.015661i HG(4,0) 0.105540-0.014269i
HG(5,1) 0.005574+0.027560i HG(0,9) 0.016402-0.104016i HG(3,3)
0.006984+0.025667i HG(0,3) 0.092075-0.035122i HG(8,1) 0.022251-0.014360i
HG(5,2) -0.092037+0.034858i HG(1,4) -0.018101-0.014466i HG(2,2)
0.089174-0.014776i HG(11,0) 0.020614-0.010033i HG(2,5) 0.079592-0.040046i
HG(7,3) 0.017058-0.014068i HG(9,0) -0.015792+0.087087i HG(1,10)
0.015733-0.000945i HG(3,0) -0.066284+0.043780i HG(1,9) 0.009205-0.011677i
HG(3,4) -0.074522+0.020353i HG(5,5) 0.009195-0.005310i HG(10,0)
0.072191-0.022591i HG(3,7) 0.008557-0.001641i HG(2,8) 0.062475-0.038479i
HG(3,1) 0.007785-0.001349i HG(4,6) 0.062011-0.034334i HG(2,9)
-0.002583-0.007210i HG(5,0) -0.049725-0.043344i HG(6,5) 0.004232-0.005767i
HG(6,4) 0.059852-0.026250i HG(1,1) 0.006285-0.001269i HG(4,3)
0.060351-0.023438i HG(3,8) 0.005092+0.001007i HG(1,6) -0.062262+0.016535i
HG(5,6) 0.003077+0.002502i HG(2,7) 0.020276-0.058881i HG(10,1)
0.000208+0.003889i HG(6,1) 0.058091-0.010760i HG(1,3) -0.003512+0.000830i
HG(8,2) 0.052511-0.025760i HG(4,7) -0.000359-0.003209i HG(2,1)
0.049257-0.030469i HG(9,2) -0.000481-0.003019i HG(0,5) 0.045250-0.033775i
HG(8,3) 0.001417+0.001375i HG(7,2) -0.020618+0.052083i HG(7,4)
-0.000059-0.000506i

##  Appendix E The parameters of the MEM beam for the Nearly-Gaussian beam II

The waist of the basic HG modes of the MEM beam for the Nearly-Gaussian beam
II in Fig. 4 is $w_{0}=$399.465\text{\,}\mu\mathrm{m}$$. The corresponding
complex coefficients for different HG modes are shown in Appendix E. The HG
modes in Appendix E are ordered in the absolute amplitude of complex
coefficient $a_{mn}$ decreasingly.

complex coefficients for different HG modes of Nearly-Gaussian beam II  mode
HG(m,n) complex coefficient $a_{mn}$ mode HG(m,n) complex coefficient $a_{mn}$
HG(0,0) 1.000000+0.000000i HG(8,0) -0.019123+0.021579i HG(2,0)
0.278731+0.219901i HG(1,6) -0.003373+0.028556i HG(0,2) -0.278731-0.219901i
HG(1,8) 0.025500-0.011701i HG(7,0) 0.091524+0.172152i HG(1,5)
0.011388-0.025343i HG(5,0) 0.074602-0.159708i HG(8,1) 0.026540-0.005864i
HG(0,4) 0.045154+0.159202i HG(6,4) 0.009602-0.024680i HG(2,2)
-0.077211-0.129553i HG(4,6) 0.023531+0.011102i HG(3,0) -0.147112-0.019083i
HG(6,1) -0.024231-0.004293i HG(4,0) -0.102501+0.088332i HG(3,5)
0.023611+0.005056i HG(9,0) -0.109524-0.077033i HG(0,5) -0.010220-0.020312i
HG(6,0) 0.009151-0.097677i HG(5,6) -0.007497-0.021052i HG(7,2)
0.011396-0.087076i HG(2,8) -0.021737+0.002029i HG(1,1) -0.046646-0.072063i
HG(3,6) -0.010431+0.017490i HG(4,2) 0.068784-0.051119i HG(0,11)
-0.007996-0.016383i HG(5,2) -0.062910+0.053035i HG(5,5) -0.016094-0.005983i
HG(2,4) 0.011725+0.073327i HG(9,1) 0.001334+0.016860i HG(0,6)
0.035398-0.063483i HG(6,3) 0.012951+0.010398i HG(11,0) 0.036618+0.061069i
HG(7,3) -0.002000+0.015624i HG(3,2) 0.045392+0.043331i HG(4,1)
0.012962+0.007516i HG(1,3) 0.013945+0.055936i HG(1,10) -0.010509-0.010347i
HG(4,4) -0.049109+0.017232i HG(2,7) -0.013172-0.002769i HG(1,2)
0.036656+0.036444i HG(2,5) 0.001213-0.013254i HG(5,4) 0.049164+0.001326i
HG(4,3) -0.008941-0.009230i HG(6,2) -0.001349+0.044950i HG(10,1)
-0.012455+0.002736i HG(2,6) 0.021935-0.038604i HG(2,1) 0.012290+0.002338i
HG(3,4) -0.002739-0.040826i HG(2,3) 0.011532+0.001631i HG(9,2)
0.001930+0.040420i HG(1,7) -0.008886+0.005939i HG(3,1) 0.021898-0.029556i
HG(8,3) -0.007374-0.006575i HG(0,7) -0.016107+0.032728i HG(2,9)
0.000048+0.009559i HG(0,9) 0.036413-0.001986i HG(3,7) 0.001158-0.009293i
HG(1,4) -0.019041-0.029166i HG(0,10) 0.003370+0.008609i HG(5,1)
0.002507+0.033980i HG(4,5) 0.006450+0.005334i HG(3,3) -0.028347+0.016449i
HG(8,2) -0.003821-0.004496i HG(7,4) -0.030367+0.009496i HG(3,8)
0.004060+0.001224i HG(10,0) 0.030037+0.008183i HG(6,5) 0.000037-0.003933i
HG(5,3) 0.018109-0.024213i HG(0,3) 0.003511+0.001090i HG(7,1)
-0.019281-0.022492i HG(1,9) 0.003354-0.000715i HG(0,8) -0.028999-0.001569i
HG(4,7) -0.000144-0.003408i

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