# Freely Decaying Saffman Turbulence Experimentally Generated by Magnetic
Stirrers

Jean-Baptiste Gorce<EMAIL_ADDRESS>Eric Falcon  Université
Paris Cité, CNRS, MSC Laboratory, UMR 7057, F-75 013 Paris, France

###### Abstract

We investigate experimentally the decay of three-dimensional hydrodynamic
turbulence, initially generated by the erratic motions of centimeter-size
magnetic stirrers in a closed container. Such zero-mean-flow homogeneous
isotropic turbulence is well suited to test Saffman’s model and Batchelor’s
model of freely decaying turbulence. Here, we report a consistent set of
experimental measurements (temporal decay of the turbulent kinetic energy, of
the energy dissipation rate, and growth of the integral scale) strongly
supporting the Saffman model. We also measure the conservation of the Saffman
invariant at early times of the decay and show that the energy spectrum scales
as $k^{2}$ at large scales and keeps its self-similar shape during the decay.
This letter thus presents the first experimental evidence of the validity of
the connection between the Saffman invariant and the $k^{2}$-energy spectrum
of the large scales. The final decay regime closely corresponds to Saffman’s
model when the container size is sufficiently large.

## Introduction.—

The decay of three-dimensional (3D) turbulent flows has been extensively
investigated to comprehend the energy transfer and the dynamics of the large
scales, the scales larger than the forcing scale Davidson . Understanding the
decay rate of turbulent kinetic energy is important for fundamental theories,
numerical simulations of turbulence and applications such as weather
forecasting or energy harvesting. However, the physical mechanisms that
control the decay rate of fully developed homogeneous turbulence are not
clearly identified Davidson .

Currently, the Batchelor model Kolmogorov ; Batchelor56  and Saffman model
Saffman67  have competing hypotheses to describe the decay of homogeneous
turbulence. Both models assume distinct invariants depending on how the
turbulence is initially generated, and this distinction is reflected in the
scaling of the energy spectrum at large scales. Specifically, a turbulent flow
with significant linear momentum possesses an energy spectrum at large scales
given by $E(k)\sim k^{2}$ (Saffman) Saffman67 . Conversely, a turbulent flow
initially generated by a strong angular impulse and a negligible linear
impulse exhibits a $E(k)\sim k^{4}$ energy spectrum at large scales
(Batchelor) Batchelor56 .

Both types of turbulence can be generated in direct numerical simulations
Davidson ; Lesieur ; IshidaJFM2006 ; DavidsonJFM2012 ; YoshimatsuPRF2019 ;
Anaspof2020 , and this raises questions about how the initial conditions or
energy injection methods control the decay of turbulent flows. Direct
numerical simulations studies on freely decaying turbulence impose the
spectrum at large scales using a Gaussian process to inject energy Kaneda04 ,
while the small scales are not turbulent and do not exhibit a $k^{-5/3}$
power-law spectrum.

Experimental open systems, such as grid turbulence, can reach a Reynolds
number up to $5\times 10^{6}$ Sinhuber2015  and are plausible candidates to
measure the decay of turbulence. However, they possess a mean flow and
different decay rates were then reported using passive grids Batchelor56 ;
Comte-Bellot66 ; Mohamed90 ; Krogstad2010 ; Sinhuber2015 , fractal grids with
multiscale grids Hurst2007 ; DavidsonJFM2011 ; Valente2011 ; Valente2012  or
active grids Burattini2005 ; Mazellier2008 ; Thormann2014 . On the other hand,
there exist complementary laboratory experiments in closed systems (where
fans, loudspeakers, jets, or rotating elements energize the fluid) generating
zero-mean-flow homogeneous isotropic turbulence (HIT) to study the decay of
turbulence Nezami2023 . However, the decay rate in such closed systems is
influenced by the different degrees of isotropy, the asymmetry of the forcing,
or secondary large-scale flows Nezami2023 . Indeed, the influence of a mean
flow or secondary flows affects the energy budget and the time dependence of
the turbulent kinetic energy, which stresses why isotropy is crucial to test
the decay law Moisy2011 . More direct evidence in zero-mean-flow HIT
experimental setups is thus required to confirm Saffman’s or Batchelor’s model
and to clarify the connection between the large-scale energy spectrum and the
invariants of freely decaying turbulence.

Here, we initially generate 3D hydrodynamic turbulence using centimeter-size
magnetic stirrers immersed in a large liquid reservoir and we then halt the
forcing to study freely decaying turbulence. The advantage of such volume
forcing is to generate sufficient zero-mean-flow HIT required to compare
Saffman’s model and Batchelor’s model of freely decaying turbulence. Using
this technique, we report a consistent set of experimental observations
(kinetic energy, dissipation rate, and integral scale) robustly supporting the
Saffman model. We also measure the conservation of the Saffman invariant at
early times of the decay. The energy spectrum scales as $k^{2}$ at large
scales and conserves a self-similar shape during the decay.

## Theoretical backgrounds.—

Assuming that the energy spectrum $E(k,t)$ is analytic at $k=0$, a Taylor
expansion at small $kr$ (large scales) shows the following leading terms
DavidsonJFM2011

$\displaystyle E(k,t)=\frac{Lk^{2}}{4\pi^{2}}+\frac{Ik^{4}}{24\pi^{2}}+...$
(1)

with
$L=\int_{0}^{\infty}\left<\mathbf{u}\left(\mathbf{x},t\right)\cdot\mathbf{u}\left(\mathbf{x+r},t\right)\right>dr$
is Saffman’s integral, a measure of the linear momentum held in the turbulence
Davidson2011 ,
$I=-\int_{0}^{\infty}\left<\mathbf{u}\left(\mathbf{x},t\right)\cdot\mathbf{u}\left(\mathbf{x+r},t\right)\right>r^{2}dr$
is Loitsyansky’s integral, suggested to be related to the angular momentum
Landau  and
$\left<\mathbf{u}\left(\mathbf{x},t\right)\cdot\mathbf{u}\left(\mathbf{x+r},t\right)\right>$
the autocorrelation function of the velocity field $\mathbf{u}$ Davidson ;
Davidson2011 . In fully developed freely decaying HIT, $L\sim u^{2}l^{3}$ with
$l$ the integral scale defined as $l=\int_{0}^{\infty}f(r,t)dr$, where
$f(r,t)$ is the longitudinal velocity autocorrelation function. Unlike $L$,
the integral $I$ is not, in general, an invariant during the initial decay
Davidson ; Batchelor56 ; Proudman .

The decay rate of the squared velocity fluctuations
$u^{2}=\left<\mathbf{u}^{2}\right>/3$ can be evaluated by assuming that
$du^{2}/dt$ is equal to minus the dissipation rate $-\epsilon$ Kolmogorov

$\frac{du^{2}}{dt}=-\epsilon=-C\frac{u^{3}}{l}$ (2)

with $C$ a constant of order unity, which depends on the Taylor Reynolds
number and the large-scale forcing procedures Sreenivasan ; Lohse ;
Sreenivasan1998 ; Kaneda2003 ; Vassilicos2015 . Using the invariant
$u^{2}l^{3}$ (Saffman) or $u^{2}l^{5}$ (Batchelor), the time dependence of
$u^{2}$, $l$, and $\epsilon$ can be derived Davidson ; Davidson2011 , as
summarized in Table 1. The decay of the kinetic energy during the final period
of decay is also shown in Table 1.

Model | Saffman | Batchelor
---|---|---
Large-scale spectrum | $E(k)\sim k^{2}$ | $E(k)\sim k^{4}$
Initial decay |  |
Invariant | $L\sim u^{2}l^{3}$ | $I\sim u^{2}l^{5}$
$u^{2}/u_{0}^{2}$ | $\left(1+at\right)^{-6/5}$ | $\left(1+bt\right)^{-10/7}$
$l/l_{0}$ | $\left(1+at\right)^{2/5}$ | $\left(1+bt\right)^{2/7}$
$\epsilon/\epsilon_{0}$ | $\left(1+at\right)^{-11/5}$ | $\left(1+bt\right)^{-17/7}$
Final decay |  |
$u^{2}$ | $\left(t-t_{*}\right)^{-3/2}$ | $\left(t-t_{*}\right)^{-5/2}$
Table 1: Time evolution of $u^{2}$, $l$, and $\epsilon$ during the initial
decay and of $u^{2}$ during the final decay depending on the initial
conditions of the turbulent flow. The large-scale energy spectrum $E(k)\sim
k^{2}$ corresponds to Saffman’s model and $E(k)\sim k^{4}$ corresponds to
Batchelor’s model. The values of the constants are
$a=\frac{5}{6}C\frac{u_{0}}{l_{0}}$ and $b=\frac{7}{10}C\frac{u_{0}}{l_{0}}$.
The initial values are indexed with 0: $u_{0}$, $l_{0}$ and $\epsilon_{0}$.

## Experimental setup.—

Experiments are carried out in two different fluid square containers sealed by
a transparent lid. The dimensions are $11\times 11\times 8$ cm3 (small tank)
and $33\times 33\times 20$ cm3 (large tank) (see the schematics in the
Supplemental Material supmatt ). The choice of these varying sizes allows for
assessing finite-size effects in the experimental observations. In the small
tank, measurements are taken using two different liquids: either water or a
lower-viscosity liquid (Novec) while exclusively water is used for
measurements in the large tank. Both fluids are seeded with hollow glass
sphere fluid tracers (10 $\upmu$m, concentration of 0.21 ppm) illuminated by a
horizontal laser sheet, and a high-speed camera (Phantom v1840) records high-
resolution movies ($2048\times 1952$ pixels2) at a range of speeds 100–400
fps. Energy is transferred into the fluid by the continuous erratic motions of
$N$ magnetic stirrers (1 cm in size) driven by a monochromatic vertical
magnetic field of frequency $F$ Falcon2013 ; Falcon2017 ; Gorce2023 , which
generate a turbulent flow Cazaubiel2021 ; Gorce2022 .

The control parameters in the small tank are the number of magnetic stirrers
$N=50$, the frequency of the oscillating magnetic field $F=50$ Hz, and the
magnetic field intensity $B=240$ G and correspond to the maximal values of
this system (see the illustrative movie in the Supplemental Material supmatt
). The typical rms velocity of the magnetic stirrers in water is 20 cm/s
Gorce2023 . The control parameters in the large tank are $N=450$, $F=20$ Hz,
and $B=360$ G.

At $t=0$, turning off the magnetic field stops the energy injection and
settles the stirrers at the container’s bottom. During this transient regime
of turbulent decay, a nonintrusive particle image velocimetry (PIV) technique
adrian1991particle  using the PIVlab algorithm Thielicke2014  measures the
fluid velocity field in the $xy$ horizontal plane. For the small tank, the
initial value of the standard deviation of the fluid velocity is equal to
$u_{0}=6.6$ cm/s, giving the initial Reynolds number
$\mathrm{Re_{0}}=u_{0}l_{0}/\nu_{w}=3000$, with $l_{0}=5$ cm the initial
integral length scale and $\nu_{w}=10^{-6}$ m2/s the kinematic viscosity of
water.

Figure 1: Time evolution of Saffman invariant $u^{2}l^{3}$ using water as
working fluid. The solid line represents the mean value of the invariant up to
$t_{1}=0.54$ s. Inset: $l/l_{0}$ as a function of $u_{0}/u$. The equation of
the solid line is $y=\left(u_{0}/u\right)^{2/3}$ (Saffman) and the dashed line
is $y=\left(u_{0}/u\right)^{2/5}$ (Batchelor). The black arrow represents the
direction of time and the dashed line gives the time $t_{1}$ after which
$u^{2}l^{3}$ decreases significantly.

## Mean-flow free, homogeneity, and isotropy.—

Using the horizontal velocity fluctuations $u_{x}$ and $u_{y}$, the structure
functions $S_{2}^{u_{x}}(r)=\langle[u_{x}(x+r)-u_{x}(x)]^{2}\rangle_{x}$ and
$S_{2}^{u_{y}}(r)$ are measured nearly identical, illustrating the homogeneity
and isotropy of the velocity field during the decay in the small tank (see
Supplemental Material supmatt ). The isotropy coefficient is also measured
using the ratio of the standard deviations.$\sigma_{u_{x}}/\sigma_{u_{y}}$ is
equal to $1\pm 0.004$ on average during the decay. The ratio of the mean
velocity and standard deviation, $\left<u_{x}\right>/\sigma_{u_{x}}$ and
$\left<u_{y}\right>/\sigma_{u_{y}}$, are 2.2% and 7%, respectively (see
Supplemental Material supmatt ), confirming the isotropy, the absence of mean
and secondary flows.

Figure 2: Decay of the squared velocity fluctuations $u^{2}$ as a function of
the rescaled time $1+at$ (water). The solid line corresponds to a power law
defined as $\left(1+at\right)^{-6/5}$ (Saffman) and the dashed line represents
the power law $\left(1+at\right)^{-10/7}$ (Batchelor). Inset: time evolution
of the integral scale $l$. The solid line represents the power law
$\left(1+at\right)^{2/5}$ (Saffman) and the dashed line
$\left(1+at\right)^{2/7}$ (Batchelor).

## Initial decay.—

The initial decay in the small tank is evaluated using exclusively water.PIV
measurements of the horizontal $u_{x}$ and vertical velocity $u_{z}$ between
$z=6$ to $8$ cm confirm the turbulent decay is not affected by the downward
motion of the stirrers (see Supplemental Material supmatt ). Figure 1 shows
that the quantity $u^{2}l^{3}$ is invariant at the beginning of the decay
until it decreases after a time $t_{1}=0.54$ s. This illustrates the
invariance of Saffman’s integral $L\sim u^{2}l^{3}$ and the conservation of
linear momentum during the initial decay ($t<t_{1}$). The present measurement
supports the hypothesis that the magnetic stirrers inject strong linear
momentum into the turbulent eddies ($L>0$), which is also endorsed by the
comparison of the time evolutions of the quantities $u^{2}l^{3}$ and
$u^{2}l^{5}$ shown in Supp. Mat. supmatt . The inset of Fig. 1 also
illustrates a power-law relationship between $1/u$ and $l$ with a 2/3 slope
consistent with Saffman’s theory, as indicated by the solid line.

Figure 3: Decay of the energy dissipation rate $\epsilon$ as a function of the
rescaled time $1+at$, with water as working fluid. The solid line represents
$\left(1+at\right)^{-11/5}$ (Saffman) and the dashed line
$\left(1+at\right)^{-17/7}$ (Batchelor). The initial value of the dissipation
rate is $\epsilon_{0}=2.1\times 10^{-3}$ m2/s3. Inset: time evolution of the
constant $C$ measured from the ratio $\epsilon l/u^{3}$. The solid line
represents the mean value of $C$ for $t\leq t_{1}$.

The measurements shown in Fig. 1 suggest a potential Saffman turbulence
scenario (second column in Table 1) in which the turbulent kinetic energy
should decay as $u^{2}/u_{0}^{2}=\left(1+at\right)^{-6/5}$ and the integral
length scale increases as $l/l_{0}=\left(1+at\right)^{2/5}$, with
$a=5Cu_{0}/(6l_{0})$. The value $a=2.7$ s-1 is inferred from the initial
values of $u_{0}$ and $l_{0}$, and the constant $C=0.37\pm 0.02$ measured from
Eq. (2). A correct definition of this value is essential for accurately
assessing the time dependence of $u$ and $l$ during the decay Mohamed90 .

Figure 2 shows the decay of $u^{2}/u_{0}^{2}$ as a function of the rescaled
time $1+at$. It confirms the power-law relationship between these two
quantities and the agreement with Saffman’s model for $t\leq t_{1}$. The inset
of Fig. 2 illustrates that the integral length scale $l$ increases during the
decay and then saturates at $\left(1+at\right)\approx 6$ (i.e., $t\approx
1.85$ s). For $t\leq t_{1}$, $l/l_{0}$ is well fitted by the solid line given
by Saffman’s model and depicts a stronger increase in $l$ than in Batchelor’s
model. Deviations of $u^{2}$ and $l$ from the Saffman laws (solid lines) are
observed after a time $1+at_{1}=2.4$ because the size of the biggest eddies
[$l(t_{1})=7$ cm] becomes comparable with the size of the container.

The rate at which the kinetic energy is dissipated is computed from the
expression $\epsilon=15\nu\langle\left(\partial u_{x}/\partial
x\right)^{2}\rangle_{x,y}$, which is derived assuming HIT Wang2021 . The
measured initial dissipation rate is equal to $\epsilon_{0}=2.1\times 10^{-3}$
m2/s3. Figure 3 shows that the decrease of $\epsilon$ is in good agreement
with Saffman’s model. The measurements are very well fitted by
$\left(1+at\right)^{-11/5}$, which is represented by the solid line in Fig. 3.
The inset of Fig. 3 represents the time evolution of the constant $C$ given by
Eq. (2). This illustrates that $C$ is approximately constant up to $t=t_{1}$,
suggesting that the velocity field is not fully turbulent after $t_{1}$ and
that different physical mechanisms dissipate the turbulent kinetic energy of
the liquid such as dissipation at the tank boundaries.

Figure 4: Decay of the turbulent kinetic energy in the small tank with two
different fluids. The blue circles correspond to the measurement performed
with water and the green squares correspond to Novec. The solid lines
represent a $t^{-1}$ power law. Inset: measurements performed in the large
reservoir filled with water. The solid line represents a $t^{-1.25}$ power
law.

## Final decay.—

After $t_{1}$, the nonlinear inertial terms in the equations of motion are
supposedly negligible and the dissipation of the turbulent kinetic energy
solely depends on the viscosity $\nu$. The evolution of the turbulent kinetic
energy during this final decay period can be derived from the initial large-
scale spectrum (see Supplemental Material supmatt ). As summarized in Table 1,
the expression is given by either $u^{2}\sim\left(t-t_{*}\right)^{-3/2}$ for
$E(k)\sim k^{2}$ Saffman67  or $u^{2}\sim\left(t-t_{*}\right)^{-5/2}$ for
$E(k)\sim k^{4}$ Batchelor53 , where $t_{*}$ denotes some instant of time
inside the final period Batchelor53 . These power laws are derived under the
assumptions that $\left(t-t_{*}\right)\rightarrow\infty$, which is challenging
to achieve in experimental systems during the final decay stage. In addition,
Ref. Skrbek2000  pointed out that the value of the power-law exponent $\alpha$
in $\left(t-t_{*}\right)^{-\alpha}$ is highly sensitive to the choice of the
virtual time parameter $t_{*}$. Consequently, we have chosen to directly fit
the experimental data using a power-law model without introducing a virtual
time origin $t_{*}$.

We conducted experiments in the small tank using two fluids (water or Novec)
with different densities and viscosities to explore how these fluids dissipate
turbulent kinetic energy during the final decay. The kinematic viscosity of
Novec 7100 is $\nu_{n}=0.4\times 10^{-6}$ m2/s and its density is
$\rho_{n}=1.5\times 10^{3}$ kg/m3 novec . Figure 4 illustrates the decays of
the turbulent kinetic energy with water (circles) and Novec (squares) that are
both very well fitted by a $t^{-1}$ power law. The exponents of the power laws
are independent of the kinematic viscosity $\nu$, which is consistent with the
theoretical derivation (see Supp. Mat. supmatt ). The deviation of the
exponent from the value -3/2 derived in Saffman’s model is likely due to the
size of the biggest eddies [$l(t_{1})=7$ cm] becoming comparable with the size
of the container. This effect is known to alter the power-law exponent of the
decay Skrbek2000 ; Thornber2016 ; Meldi2017 . Additionally, finite Reynolds
number effects contribute to this deviation Anaspof2020 .

To reduce the finite-size effects of the small tank and the dissipation at its
boundaries, we also conducted experiments within the large tank. The inset of
Fig. 4 shows that a $t^{-1.25}$ power law is observed for 1 order of
magnitude. This supports the fact that the finite-size effects control the
decay rate during the final period of decay and the time power-law exponent
becomes closer to -3/2 (Saffman’s model) in the large tank experiment. Note
that the initial decay is not observed in the large tank because the initial
Reynolds number is too small
($\mathrm{Re_{0}^{\prime}}=u_{0}^{\prime}l_{0}/\nu_{w}=650$, with
$u_{0}^{\prime}=1.3$ cm/s). Indeed, Fig. 5 illustrates that the $k^{-5/3}$
power spectrum is no longer observed after only 0.01 s, which is clearly
insufficient to resolve correctly the initial decay.

Figure 5: Decay of the energy spectrum in the large reservoir. The vertical
dashed line corresponds to the initial inverse integral length $1/l_{0}$
separating the large and small scales. Here, $t=0,0.09,3.66,5.96,15.83,42.01$,
and $104.99$ s.

## Energy spectrum.—

In the absence of nonlinear transfer of energy across scales, Lin’s equation,
given by $\partial E(k,t)/\partial t\sim-2\nu k^{2}E(k,t)$, implies that the
expected $k^{2}$ energy spectrum at large scales should persist over time
throughout the decay. Measurements performed in the large tank confirm the
conservation of the $k^{2}$ power law during the final decay stage, whereas
the smaller scales lose their turbulent characteristics and exhibit a steeper
power-law trend (Fig. 5). These observations align with the idea that
viscosity dissipates the excess energy during the final decay and suggest that
Saffman turbulence is observed here.

## Conclusion.—

We report on the freely decaying 3D turbulence, initially generated by the
erratic motions of centimeter-size magnetic stirrers in a closed experimental
setup. Such isotropic, mean-flow-free turbulence is well suited to compare
Saffman and Batchelor models of freely decaying turbulence. Our experimental
measurements (temporal decay of the turbulent energy kinetic, of the energy
dissipation rate, and growth of the integral scale) robustly support Saffman
model. Saffman invariant is also well conserved at early times of the decay.
The energy spectrum scales as $k^{2}$ at large scales and conserves a self-
similar shape during the decay. This letter thus presents the first
experimental evidence of the connection between Saffman invariant $L\sim
u^{2}l^{3}$ and the large-scale energy spectrum in $k^{2}$. The final decay is
also reported in two different-size experimental systems. All these results
support the existence of freely decaying Saffman turbulence involving
turbulent eddies with a significant linear momentum input. Our results could
be applied to physical, geophysical, or industrial turbulent flows with a
finite mean flow and are of primary importance.

###### Acknowledgements.

We thank A. Di Palma and Y. Le Goas, for technical help. This work was
supported by the French National Research Agency (ANR LASCATURB project No.
ANR-23-CE30-0043-02) and by the Simons Foundation MPS No651463-Wave
Turbulence.

## References

  * (1) P. A. Davidson Turbulence (Oxford University Press, New-York, 2nd ed., 2015).
  * (2) A. N. Kolmogorov, On the degeneration of isotropic turbulence in an incompressible viscous fluid, Dokl. Akad. Nauk SSSR 31, 538 (1941).
  * (3) G. K. Batchelor and I. Proudman, The large-scale structure of homogeneous turbulence, Phil. Trans. R. Soc. A 248, 369 (1956).
  * (4) P. G. Saffman, The large-scale structure of homogeneous turbulence, J. Fluid Mech. 27, 581 (1967).
  * (5) M. Lesieur Turbulence in fluids (Martinus Nijhoff Publishers, Dordrecht, 1987) pp. 205-211.
  * (6) T. Ishida, P. A. Davidson and Y Kaneda, On the decay of isotropic turbulence, J. Fluid Mech. 564, 455 (2006).
  * (7) P. A. Davidson, N. Okamoto, and Y. Kaneda, On freely decaying, anisotropic, axisymmetric Saffman turbulence, J. Fluid Mech. 706, 150 (2012).
  * (8) K. Yoshimatsu and Y. Kaneda, No return to reflection symmetry in freely decaying homogeneous turbulence, Phys. Rev. Fluids 4, 024611 (2019).
  * (9) M. Anas, P. Joshi, and M. K. Verma, Freely decaying turbulence in a finite domain at finite Reynolds number, Phys. Fluids 32, 095109 (2020).
  * (10) Y. Kaneda, T. Ishihara, M. Yokokawa, K. Itakura and A. Uno, High-resolution direct numerical simulation of turbulence — spectra of fourth-order velocity moments, in IUTAM Symposium on Reynolds Number Scaling in Turbulent Flow. Fluid Mechanics and its Applications, edited by A.J. Smits, (Springer, Dordrecht, 2004), Vol. 74. ( pp. 155-162). 10.1007/978-94-007-0997-3_27.
  * (11) M. Sinhuber, E. Bodenschatz, and G. P. Bewley, Decay of turbulence at high Reynolds numbers, Phys. Rev. Lett. 114, 034501 (2015).
  * (12) G. Comte-Bellot and S. Corrsin, The use of a contraction to improve the isotropy of grid-generated turbulence, J. Fluid Mech. 25, 657 (1966).
  * (13) M. S. Mohamed and J. C. LaRue, The decay power-law in grid-generated turbulence, J. Fluid Mech. 219, 195 (1990).
  * (14) P. Å. Krogstad and P. A. Davidson, Is grid turbulence Saffman turbulence?, J. Fluid Mech. 642, 373 (2010).
  * (15) D. Hurst and J. C. Vassilicos, Scalings and decay of fractal-generated turbulence, Phys. Fluids 19, 035103 (2007).
  * (16) P. Å. Krogstad and P. A. Davidson, Freely decaying, homogeneous turbulence generated by multi-scale grids, J. Fluid Mech. 680, 417 (2011).
  * (17) P. C. Valente and J. C. Vassilicos, The decay of turbulence generated by a class of multiscale grids, J. Fluid Mech. 687, 300 (2011).
  * (18) P. C. Valente and J. C. Vassilicos, Dependence of decaying homogeneous isotropic turbulence on inflow conditions, Phys. Lett. A 376, 510 (2012).
  * (19) P. Burattini, P. Lavoie, and R. Antonia, On the normalized turbulent energy dissipation rate, Phys. Fluids 17, 98103 (2005).
  * (20) N. Mazellier and J. C. Vassilicos, The turbulence dissipation constant is not universal because of its universal dependence on large-scale flow topology, Phys. Fluids 20, 015101 (2008).
  * (21) A. Thormann and C. Meneveau, Decay of homogeneous, nearly isotropic turbulence behind active fractal grids, Phys. Fluids 26, 025112 (2014).
  * (22) A. G. Nezami, M. Byron, and B. A. Johnson, Laboratory generation of zero-mean-flow homogeneous isotropic turbulence: Non-grid approaches, Flow 3, E42 (2023) and references therein.
  * (23) F. Moisy, C. Morize, M. Rabaud, and J. Sommeria, Decay laws, anisotropy and cyclone–anticyclone asymmetry in decaying rotating turbulence, J. Fluid Mech. 666, 5 (2011); C. Morize, F. Moisy, and M. Rabaud, Decaying grid-generated turbulence in a rotating tank, Phys. Fluids 17, 095105 (2005).
  * (24) P. A. Davidson, The minimum energy decay rate in quasi-isotropic grid turbulence, Phys. Fluids 23, 085108 (2011).
  * (25) $I$ is only a conserved quantity for a fast enough decay of $f(r)$ when $r\rightarrow 0$, and thus a direct consequence of the conservation of angular momentum.
  * (26) I. Proudman and W. H. Reid, On the decay of a normally distributed and homogeneous turbulent velocity field, Phil. Trans. R. Soc. A 247, 163 (1954).
  * (27) K. R. Sreenivasan, On the scaling of the turbulence energy-dissipation rate, Phys. Fluids 27, 1048 (1984).
  * (28) D. Lohse, Crossover from high to low Reynolds number turbulence, Phys. Rev. Lett. 73, 3223 (1994).
  * (29) K. R. Sreenivasan, An update on the energy dissipation rate in isotropic turbulence, Phys. Fluids 10, 528 (1998).
  * (30) Y. Kaneda, T. Ishihara, M. Yokokawa, K. I. Itakura, and A. Uno, Energy dissipation rate and energy spectrum in high-resolution direct numerical simulations of turbulence in a periodic box, Phys. Fluids 15, L21 (2003).
  * (31) J. C. Vassilicos, Dissipation in turbulent flows, Annu. Rev. Fluid Mech. 47, 95 (2015) and references therein.
  * (32) See Supplemental Material at http://link.aps.org/… for further data analyses: (i) movies, (ii) a comparison between Saffman and Batchelor turbulence during the initial decay, (iii) experimental setup details and additional data supporting the isotropy of the velocity field during the decay, and (iv) the derivation of the final decay prediction.
  * (33) E. Falcon, J.-C. Bacri, and C. Laroche, Equation of state of a granular gas homogeneously driven by particle rotations, Europhys. Lett. 103, 64004 (2013).
  * (34) E. Falcon, J.-C. Bacri, and C. Laroche, Dissipated power within a turbulent flow forced homogeneously by magnetic particles, Phys. Rev. Fluids 2, 102601(R) (2017).
  * (35) J.-B. Gorce and E. Falcon, Statistics of a two-dimensional immersed granular gas magnetically forced in volume, Phys. Rev. E 107, 034903 (2023).
  * (36) A. Cazaubiel, J.-B. Gorce, J.-C. Bacri, M. Berhanu, C. Laroche, and E. Falcon, Three-dimensional turbulence generated homogeneously by magnetic particles, Phys. Rev. Fluids 6, L112601 (2021).
  * (37) J.-B. Gorce and E. Falcon, Statistical equilibrium of large scales in three-dimensional hydrodynamic turbulence, Phys. Rev. Lett. 129, 054501 (2022).
  * (38) R. J. Adrian, Particle-imaging techniques for experimental fluid mechanics, Annu. Rev. Fluid Mech. 23, 261 (1991).
  * (39) W. Thielicke and E. Stamhuis, PIVlab towards user-friendly, affordable and accurate digital particle image velocimetry in MATLAB, J. Open Res. Software 2, 1 (2014).
  * (40) G. Wang, F. Yang, K. Wu, Y. Ma, C. Peng, T. Liu, and L. P. Wang, Estimation of the dissipation rate of turbulent kinetic energy: A review, Chem. Eng. Sci. 229, 116133 (2021).
  * (41) G. K. Batchelor, The Theory of Homogeneous Turbulence (Cambridge University Press, New York, 1953).
  * (42) L. Skrbek and S. R. Stalp, On the decay of homogeneous isotropic turbulence, Phys. Fluids 12, 1997 (2000).
  * (43) Values provided at $20^{\circ}$C by the manufacturer 3M.
  * (44) B. Thornber, Impact of domain size and statistical errors in simulations of homogeneous decaying turbulence and the Richtmyer-Meshkov instability, Phys. Fluids 28, 045106 (2016).
  * (45) M. Meldi and P. Sagaut, Turbulence in a box: Quantification of large-scale resolution effects in isotropic turbulence free decay, J. Fluid Mech. 818, 697 (2017).


# Research on fine co-focus adjustment method for segmented solar telescope

Kunyan Wang  1,2 Yichun Dai  1,3* Bin Wang  1,3 Xu Tan  1,2 Dehua Yang  4 and
Zhenyu Jin1,3  1Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, China
2University of Chinese Academy of Sciences, Beijing 100049, China
3Yunnan Key Laboratory of Solar Physics and Space Science, 650216, China
4Nanjing Institute of Astronomical Optics and Technology, Chinese Academy of
Sciences, Nanjing 210042, China<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

For segmented telescopes, achieving fine co-focus adjustment is essential for
realizing co-phase adjustment and maintenance, which involves adjusting the
millimeter-scale piston between segments to fall within the capture range of
the co-phase detection system. CGST proposes using a SHWFS for piston
detection during the co-focus adjustment stage. However, the residual piston
after adjustment exceeds the capture range of the broadband PSF phasing
algorithm$(\pm 30\mu m)$, and the multi-wavelength PSF algorithm requires even
higher precision in co-focus adjustment. To improve the co-focus adjustment
accuracy of CGST, a fine co-focus adjustment based on cross-calibration is
proposed. This method utilizes a high-precision detector to calibrate and fit
the measurements from the SHWFS, thereby reducing the impact of atmospheric
turbulence and systematic errors on piston measurement accuracy during co-
focus adjustment. Simulation results using CGST demonstrate that the proposed
method significantly enhances adjustment accuracy compared to the SHWFS
detection method. Additionally, the residual piston after fine co-focus
adjustment using this method falls within the capture range of the multi-
wavelength PSF algorithm. To verify the feasibility of this method,
experiments were conducted on an 800mm ring segmented mirror system,
successfully achieving fine co-focus adjustment where the remaining piston of
all segments fell within $\pm 15\mu m$.

##  1 Introduction

The Chinese Giant Solar Telescope (CGST) is a next-generation giant solar
telescope program jointly proposed by the Chinese solar physics community. A
significant option for CGST’s primary mirror involves employing ring-segmented
mirrors. The scientific objective of CGST is to measure the delicate
structures of magnetic and flow fields across various levels of the solar
atmosphere, as well as their high spatial and temporal resolution evolutionary
processes. For this purpose, the telescope is required to achieve co-phase
within the 1 $\mu$ m wavelength range to realize high-resolution observation
and research on the 20-km delicate structures of the solar surface[1, 2, 3].
Due to the high precision requirement for co-phasing, which is in the order of
nanometers, the capture range of the classical phasing algorithm is limited to
±30$\mu$m (Keck’s broadband PSF phasing algorithm)[4]. However, following the
initial mechanical alignment, the piston between segments may reach the
millimeter-scale. Therefore, it is necessary to adjust the large-scale piston
before performing co-phasing, referred to as co-focus adjustment. In
comparison with traditional co-focus adjustment, the paper focuses on fine co-
focus adjustment, which not only entails adjusting the millimeter-scale piston
to the capture range of the co-phase measurement system but also improving the
accuracy of the co-focus adjustment, thereby simplifying the subsequent co-
phase adjustment process.

Currently, the primary methods used for co-focus adjustment in segmented
telescopes include spherometer measurement[4], Shack-Hartmann Wavefront
Sensor(SHWFS) detection[5, 6], and interferometer measurement[7]. The 10m Keck
telescope in the United States utilized a hand-held spherometer to adjust the
large-scale piston within the capture range of the broadband PSF phasing
algorithm $(\pm 30\mu m)$[4]. Similarly, telescopes like the 9.2m SALT in
South Africa and the LAMOST in China employed SHWFS for co-focus adjustment,
where defocus measurement reflects the segment’s large-scale piston[5, 6, 8,
9]. However, the SHWFS of SALT had a practical detection accuracy of 60
$\mu$m(Root Mean Square, RMS) for large-scale piston, leading to the adoption
of spherometer measurement, achieving co-focus adjustment with an accuracy of
15 $\mu$mRMS[5]. Furthermore, interferometer measurement was proposed for co-
focus adjustment in the 9.2m HET telescope in the United States, aiming for an
accuracy of 25 $\mu$mRMS. However, due to insufficient robustness in the
observing environment, SHWFS was ultimately employed for co-focus
adjustment[7].

When the piston between segments reaches the millimeter-scale, it results in
defocus. Therefore, the piston during the co-focus stage can be approximately
obtained from the defocus measurement using a SHWFS. This method offers a more
straightforward implementation than a spherometer or interferometer and can
detect tip/tilt accurately. CGST plans to use SHWFS for piston measurement in
the co-focus stage. However, due to atmospheric turbulence and systematic
error[10, 11], the current accuracy of SHWFS measurement for detecting large-
scale piston is about 60 $\mu$mRMS (SALT), which falls short of the capture
range of typical phasing algorithm ($\pm 30\mu m$ for Keck’s broadband PSF
phasing algorithm). Additionally, the broadband PSF phasing algorithm is
cumbersome because it requires scanning at a fixed step size, and the actuator
displacements result in cumulative errors with too many scans. The co-focus
measurement of TMT favors a multi-wavelength PSF detection method, which has a
capture range of about $\pm 15\mu m$, imposing higher demands on the accuracy
of co-focus adjustment[12, 13]. This paper proposes a fine co-focus adjustment
based on cross-calibration to improve the performance of the co-focus of CGST
and facilitate the subsequent co-phase adjustment. The method employs a
detector with higher detection accuracy to calibrate and fit the measurement
results obtained from SHWFS. This approach aims to diminish the effects of
atmospheric turbulence and systematic error on the measurement accuracy of the
piston during the co-focus stage and improve the adjustment accuracy of the
co-focus. To better verify the feasibility of this method, experiments were
conducted on the 800mm ring segmented mirror system, and the adjustment
results were analyzed.

Section 2 of this paper analyzes the fine co-focus adjustment method for CGST.
It introduces the fine co-focus adjustment based on cross-calibration proposed
in this paper and conducts a simulation analysis of the potential errors in
the actual measurement. Section 3 presents the experimental results of fine
co-focus adjustment using the above method on an 800mm ring segmented mirror
system. The conclusion is to be presented in Section 4.

##  2 Analysis of fine co-focus adjustment method for CGST

An important alternative for the primary mirror of CGST is the utilization of
ring-segmented mirrors. This configuration consists of 24 segments, each of
which is an annular sector. The segments have a long base of 1040 mm, a short
base of 779 mm, and a height of 1015 mm. The primary mirror is a parabolic
reflector with a ring width of 1 m and a focal ratio of 1, as illustrated in
Fig. 1[1].

Figure 1: Schematic diagram of CGST and primary mirror

###  2.1 The principle and detection accuracy of SHWFS

The SHWFS for co-focus adjustment is placed at the exit pupil of the optical
system. Sixteen sub-apertures are planned to be allocated inside each segment
to detect tip/tilt and piston during the co-focus stage. An additional two
sub-apertures are placed at each edge for piston sensing during the co-phase
stage, resulting in a total of 432 sub-apertures, as depicted in Fig. 2(a).

For SHWFS detection, $Z_{2}$, $Z_{3}$ and $Z_{4}$ denoting tip/tilt and
defocus can typically be reconstructed using the modal method[14, 15]. During
the co-focus stage, the piston between segments is large, which can be
approximately obtained from defocus measurements. $Z_{4}$ denotes the piston
during the co-focus stage in the paper. Due to the unique structure of the
ring segmented mirrors, in this paper, the outer circle of the micro-lens
array, corresponding to each segment, is selected as the unit-orthogonal
circular domain for the Zernike polynomials, as illustrated in Fig.2(b). It
has been verified that in the annular sector region, the 2nd Zernike
polynomial is orthogonal to the 3rd Zernike polynomial, and the 3rd Zernike
polynomial is orthogonal to the 4th Zernike polynomial, can be expressed as

$\left\\{\begin{array}[]{c}\frac{1}{S}\iint_{A}(2x)\cdot(2y)d\sigma=0\\\
\frac{1}{S}\iint_{A}(2x)\cdot\sqrt{3}\left(2\left(x^{2}+y^{2}\right)-1\right)d\sigma=0\end{array}\right.$
(1)

Where A denotes the annular sector region and S is the area of the annular
sector. Therefore, using the modal wavefront reconstruction enables the
decoupling of the segment’s tip/tilt and piston.

(a)

(b)

Figure 2: The SHWFS for CGST.(a)Micro-lens array corresponding to a segment.
(b)The unit circle domain of the Zernike polynomials.

During actual detection, the accuracy of co-focus adjustment using SHWFS can
be affected by environmental factors. In the presence of atmospheric
turbulence, the mean-square value of the angle of arrival in the x or y
direction on a circular aperture with a diameter D can be mathematically
expressed as[16]

$\left\langle\alpha_{x}^{2}\right\rangle=\left\langle\alpha_{y}^{2}\right\rangle=0.170\left(\frac{\lambda}{D}\right)^{2}\left(\frac{D}{r_{0}}\right)^{5/3}$
(2)

where $\lambda$ is the wavelength and $r_{0}$ is the atmospheric coherence
length. Suppose the sub-aperture of the micro-lens array corresponds to about
9.8 cm(the value of $r_{0}$ is approximately equivalent) on the primary
mirror. In that case, the tilt error due to atmospheric turbulence is 0.55
arcsecond from Eq.(2) when $r_{0}$ is 10 cm and $\lambda$ is 650 nm. To
mitigate the impact of atmospheric turbulence, integrating multiple frames of
measurement data with successive short exposures over time can effectively
suppress the effect[17]. An atmospheric random phase screen based on the
Kolmogorov model was generated to simulate atmospheric turbulence using the
power spectral inversion method[18]. The simulation results indicate that the
accuracy of SHWFS for large-scale piston measurements is 195 $\mu$mRMS using a
single frame and improves to 64 $\mu$mRMS when using 10 frames.

Moreover, practical measurements with SHWFS may be affected by systematic
error. For instance, the temperature gradient of the air in the measurement
optical path and its slow change can introduce systematic measurement error.
Compared to the random wavefront fluctuation caused by turbulence, the spatio-
temporal frequency characteristic of the temperature gradient is lower, and
the wavefront aberration is hardly to be eliminated by smoothing[19]. This
paper proposes a fine co-focus adjustment method based on cross-calibration to
minimize the impact of various sources of uncertainty and improve the
precision of co-focus adjustment.

###  2.2 Fine co-focus adjustment based on cross-calibration

####  2.2.1 Basic Principles

The principle of cross-calibration is to calibrate the measurement results of
SHWFS using a more accurate measurement device. The cross-calibration is
performed at multiple positions before and after the initial co-focus “0
point” obtained from SHWFS. The theoretical co-focus "0-point" position can be
obtained more accurately by fitting the data. The mathematical expression of
the cross-calibration method is shown in Eq. (3).

$\hat{L}=f\left(Z_{4}\right)=\mathrm{k}*Z_{4}+b$ (3)

In Eq. (3), L represents the detection value obtained from the calibration
device with higher accuracy. $Z_{4}$ is the measured value of SHWFS,
indicating piston during the co-focus stage. The intercept b in the equation
represents the difference between the initial co-focus “0 point” obtained from
SHWFS and the more accurate theoretical co-focus “0 point” obtained from the
fine co-focus method. Therefore, after SHWFS detection and adjustment, the
segment still needs to be moved by the amount of b.

The accuracy of the fine co-focus adjustment based on cross-calibration is
determined by the solution precision of the intercept b, as indicated in Eq.
(3). The standard deviation of b, denoted as $s_{b}$, can be expressed as

$s_{b}=\frac{s_{L}}{\sqrt{n}}\sqrt{1+\frac{{\overline{Z_{4}}}^{2}}{s_{Z_{4}}{}^{2}}}$
(4)

where n represents the number of measurement points; $s_{L}$ is the root mean
square error (RMSE) of the fitting for L, given by

$s_{L}=\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}\left(L_{i}-k*Z_{4,i}-b\right)^{2}}$
(5)

$\overline{Z_{4}}$ represents the average of the measured values
$Z_{4,i}$;$s_{Z_{4}}$ is the standard deviation of the measured value
$Z_{4,i}$,given by

$s_{Z_{4}}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(Z_{4,\mathrm{i}}-\overline{Z_{4}}\right)^{2}}$
(6)

Since the measurement points are situated before and after the initial co-
focus “0 point” obtained from SHWFS,$\overline{Z_{4}}$ is small. Hence, Eq.
(4) can be approximated as

$s_{b}=\frac{s_{L}}{\sqrt{n}}$ (7)

If the detection error of the calibration device is $\Delta_{L}$, the
uncertainty of the intercept b can be expressed as shown in Eq. (8).

$U_{b}=\sqrt{U_{1}{}^{2}+{U_{2}}^{2}}=\sqrt{\left(t_{0.95}(n-2)*s_{b}\right)^{2}+\left(\Delta_{L}\right)^{2}}$
(8)

In Eq. (8), $t_{0.95}(n-2)$ represents the t-distribution factor for a
probability of 0.95 and n-2 degrees of freedom. This factor is approximately
1.8, and the exact value for different values of n can be obtained by
referring to the appropriate table.

From Eq. (8), the accuracy of the fine co-focus adjustment based on cross-
calibration is related to the number of measurement points n and the detection
error of the calibration device $\Delta_{L}$. A higher co-focus adjustment
accuracy can be achieved when n is larger and $\Delta_{L}$ is smaller. For
instance, if the detection accuracy of SHWFS is 64 $\mu$mRMS, the value of
$s_{L}$ is approximately 60 $\mu$m. When the number of measurement points n <
100, $U_{1}$ > 10 $\mu$m, $U_{b}$ is primarily determined by the magnitude of
$U_{1}$, assuming the value of $\Delta_{L}$ is a few micrometers. However,
when n surpasses 100, the influence of $U_{2}$ in Eq. (8), which represents
the detection accuracy of the calibration device, gradually becomes more
significant as n increases. Theoretically, with a sufficiently large number of
measurement points n, the highest detection accuracy of the fine co-focus
adjustment based on cross-calibration is determined by the detection accuracy
of the calibration device.

####  2.2.2 Simulation analysis

If we only consider the influence of atmospheric turbulence with an
atmospheric coherence length $r_{0}=10cm$, the simulation results of cross-
calibration for measurement points of 10 and 50 are presented in Fig. 3. In
the figure, the horizontal axis represents the $Z_{4}$ obtained from SHWFS by
using 10 frames data for wavefront reconstruction. In contrast, the vertical
axis indicates the detection value of the calibration device with a piston
moving step of 20 $\mu$m. In the analysis of this paper, the uncertainty
$U_{b}$ of the intercept b in Eq. (8) is employed to quantify the range of
residual piston after fine co-focus adjustment. It can be inferred that the
residual piston, after fine co-focus adjustment, can fall within the capture
range of the multi-wavelength PSF algorithm when the number of measurement
points n is approximately 50. Since the number of measurement points n
utilized in the simulation analysis is less than 100, the detection error of
the calibration device $\Delta_{L}$ can be neglected in the calculation. The
range of residual pistons after adjustment using different measurement points
are summarized in Table 1.The simulation results show that compared with SHWFS
detection, the fine co-focus adjustment based on cross-calibration enhances
the detection accuracy of large-scale piston. When using 10 frames data for
wavefront reconstruction and employing 50 measurement points, the range of
residual piston after the fine co-focus adjustment is $\pm$14.2 $\mu$m.

Figure 3: Simulation results of the fine co-focus adjustment based on cross-calibration (effect of atmospheric turbulence) Table 1: The range of residual piston of the fine co-focus adjustment based on cross-calibration(effect of atmospheric turbulence) Number of measurement points | 10 | 20 | 30 | 40 | 50
---|---|---|---|---|---
The range of residual piston / $\mu$m |  $\pm$29.7 |  $\pm$22.1 |  $\pm$17.1 |  $\pm$15.5 |  $\pm$14.2

Furthermore, in the presence of a systematic detection error $\sigma$, which
may arise from factors such as temperature gradient or other sources, the
result using cross-calibration is illustrated in Fig.4. The result indicates
that the segment needs to be moved by -$\sigma$ after SHWFS detection and
adjustment. Consequently, this adjustment method can also effectively mitigate
the impact of systematic errors.

Figure 4: Simulation result of the fine co-focus adjustment based on cross-
calibration(effect of systematic error)

##  3 Experimental results and analysis

###  3.1 Introduction to the experimental system

The experimental system for fine co-focus calibration is a primary mirror
composed of 8 annular sector spherical segmented mirrors, as illustrated in
Fig. 5(a). The parameters of the segmented mirror are presented in Table 2.
The co-focus detection optical path of the system is depicted in Fig. 6. In
this configuration, a light source is positioned at the aplanatic points of
the primary mirror, and SHWFS is employed to detect the tip/tilt and piston
between the segments. A ring micro-lens array is situated at the primary
mirror’s exit pupil. Within the micro-lens array, each segment has seven
internal sub-apertures and two sub-apertures at the edge, yielding 72 sub-
apertures. During the co-focus adjustment stage, the seven internal sub-
apertures are utilized to detect the segment’s tip/tilt and piston, with the
piston approximately obtained from defocus measurements. The two sub-apertures
at the edge detect the piston during the co-phase adjustment stage. Once the
co-focus adjustment is completed, a filter with a center wavelength of 636 nm
and a bandwidth of 10 nm is employed for broadband scanning to adjust the
piston, whose capture range is $\pm$15 $\mu$m. Therefore, the segment’s fine
co-focus adjustment accuracy must match the co-phase stage’s capture range.

The calibration device is the LVDT (Linear Variable Differential Transformer),
sold on the shelf with a detection accuracy of better than 3 $\mu$mRMS from
the specification. It is mounted next to the actuator and is used to measure
the actuator’s linear displacement or displacement difference, as shown in
Fig. 5(b).The actuator is screw-driven and capable of achieving high-precision
micro-motions (during the co-phase stage). However, the actuator exhibits
significant errors in the order of hundreds of micrometers for larger
displacements, such as in co-focus adjustment. The LVDT measures the linear
displacement of an object through the variation of the induced voltage in the
secondary coil and behaves with high sensitivity and good linearity. In this
paper’s experiments, the displacement of the segment is determined by the
readings of the LVDT. Additionally, the LVDT has a reference position known as
the "0 point", which indicates that when the measured object is positioned at
the center of the LVDT, the output voltage is zero.

(a)

(b)

Figure 5: 800mm ring segmented mirror system.(a)The primary mirror.(b)Actuator
and LVDT

(a)

(b)

Figure 6: Co-focus detection optical path.(a)Schematic diagram.(b)Physical image Table 2: Optical parameters of the 800mm ring segmented mirror system Parameter | Symbol | Value
---|---|---
Diameter of primary mirror | D/mm | 800
Radius of curvature | R/mm | 3200
Ring width |  $\Delta$D/mm | 120
Focal ratio | f/$\\#$ | 2
Focal length of collimator lens 1 | f1/mm | 37.5
Focal length of collimator lens 2 | f2/mm | 72.38
Focal length of imaging lens | f3/mm | 78.39
Diameter of pinhole | d1/ $\mu$m | 20
Diameter of sub-aperture | d2/ $\mu$m | 583
Focal length of micro-lens array | f4/mm | 77
Detector resolution | p*p/(pixel*pixel) | 2048*2048
Pixel size | d3*d3/( $\mu$m * $\mu$m) | 5.5*5.5

###  3.2 The detection accuracy of LVDT and SHWFS

For our SHWFS, the detection accuracy of tip/tilt is higher than 0.085
arcsecond RMS when the centroid sensing obtains sub-pixel resolution. This
level of accuracy corresponds to an actuator length of 0.08 $\mu$mRMS. The
LVDT has a detection accuracy of better than 3 $\mu$mRMS from the
specification. The actual detection accuracy of the LVDT should be determined
before it is used as the high detection means for cross-calibration of SHWFS.
Due to the high accuracy of the SHWFS in detecting tip/tilt, we utilized tip
measurements to calibrate the measurement accuracy of the LVDT. First, a
movement of 4 $\mu$m was applied to the controller of the actuator M1 to
produce a tip, while the readings of the LVDT and the detected value $Z_{3}$
of the SHWFS were recorded. Subsequently, using the relationship between
$Z_{3}$ and the displacement of actuator M1, $Z_{3}$ was converted into the
corresponding actuator displacement, considering it as a reference value. The
difference between the measured value of the LVDT and the reference value
represented the detection error of the LVDT. Fig.7 illustrates the detection
errors of the LVDT from 40 measurements, with an RMS value of 1.93 $\mu$m.
Therefore, the LVDT can be utilized to cross-calibrate the SHWFS.

Figure 7: The detection error of the LVDT

Similarly, the accuracy of LVDT is higher than SHWFS in detecting large-scale
piston, so the measurement values obtained from the LVDT can be used to
evaluate the accuracy of SHWFS in detecting large-scale piston. First, a
displacement of 50 $\mu$m was applied as the input to the controllers of three
actuators corresponding to the segment, generating a piston. Simultaneously,
the readings of the LVDT and $Z_{4}$ from the SHWFS were recorded, with the
LVDT readings as the reference value. Subsequently, using the relationship
between $Z_{4}$ and the displacement of the actuators, $Z_{4}$ was converted
into corresponding actuator displacement. The difference between the converted
value and the reference value represented the detection error of the piston.
Fig.8 illustrates the detection errors of 40 measurements of piston obtained
from the SHWFS. The RMS value of the detection errors is 20.41 $\mu$m, while
the peak-to-valley (PTV) value is 90.99 $\mu$m.

Figure 8: The detection error of the piston(SHWFS)

###  3.3 Adjustment results and analysis of fine co-focus adjustment based on
cross-calibration

Fine co-focus adjustment using LVDT and SFWFS was as follows: Firstly, the
initial co-focus "0 point" was obtained after SHWFS detection and
adjustment.To ensure that the residual piston after fine co-focus adjustment
could fall into the capture range of the broadband PSF phasing of $\pm$ 15
$\mu$m, the number of measurement points, denoted as n, could be estimated
using Eq. (8) to be 10, where the RMSE of the LVDT fitting $s_{L}$ was
approximately 15 $\mu$m. Consequently, five points were measured in 50
$\mu$m(controller input value for actuator) increments both before and after
the initial co-focus "0 point", and the corresponding LVDT readings and
$Z_{4}$ measured by SHWFS were simultaneously recorded. A least-squares
fitting was then performed on the variations in LVDT readings and the
corresponding $Z_{4}$ .Finally, the value of LVDT when $Z_{4}$=0 was
calculated, representing the additional displacement required for the segment
to obtain a more precise theoretical co-focus “0-point”. Fig. 9 illustrates
the fine co-focus adjustment results of segment 1 repeated four times. The
horizontal axis represents the detection value $Z_{4}$ of the SHWFS, while the
vertical axis represents the variation in LVDT readings. "-22$\mu$m",
"-26$\mu$m", "-15$\mu$m", and "-30$\mu$m" represent the adjustments required
for segment 1 after SHWFS detection and adjustment (results of 4 experiments).

Figure 9: Results of multiple co-focus adjustments for segment 1.

In the fine co-focus adjustment, the range of residual piston is estimated
from the uncertainty $U_{b}$ of the intercept b in Eq. (8), where the value of
$t_{0.95}(n-2)$ is determined to be 1.86 for n is 10. The standard deviation
$s_{b}$ of the intercept b, obtained by fitting the measured data of the eight
segments, is presented in Table 3. The detection error $\Delta_{L}$ of the
LVDT is 3 $\mu$mRMS, which can be disregarded in the calculation. The range of
residual pistons after the adjustment of the eight segments using the fine co-
focus adjustment based on cross-calibration are displayed in Table 4.

Table 3: Standard deviation of the intercept b obtained from real measurements of 800mm ring segmented mirror system segment | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
$s_{b}$/ $\mu$m | 6.1 | 7.0 | 4.9 | 4.6 | 7.0 | 2.7 | 3.0 | 4.7
Table 4: The range of residual piston after fine co-focus adjustment for 800mm ring segmented mirror system segment | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---
The range of residual piston / $\mu$m |  $\pm$11.3 |  $\pm$13.0 |  $\pm$9.1 |  $\pm$8.6 |  $\pm$13.0 |  $\pm$5.0 |  $\pm$5.6 |  $\pm$8.7

Based on the experimental results presented in Table 4, it can be observed
that when the number of measurement points is 10, the detection accuracy of
fine co-focus adjustment based on cross-calibration can reach 26$\mu$m(PTV),
and the residual pistons of all segments after fine co-focus adjustment fall
into the capture range of the broadband PSF phasing($\pm$15 $\mu$m).
Furthermore, these residual pistons also remain within the capture range of
the multi-wavelength PSF algorithm. As a result, the subsequent co-phase
adjustment can be carried out.

##  4 Conclusion

This paper presents the fine co-focus adjustment based on cross-calibration
with high precision for the CGST. The accuracy of this method relies on two
key factors: the number of measurement points used for cross-calibration and
the detection accuracy of the calibration device. This method effectively
mitigates the detection error caused by atmospheric turbulence and reduces the
impact of systematic error. Compared to the SHWFS detection system, the method
significantly improves the accuracy of co-focus adjustment. Moreover, when
using 50 measurement points, the residual piston is within $\pm$14.2 $\mu$m,
ensuring it falls within the capture range of the multi-wavelength PSF phasing
algorithm. As a result, this method simplifies the following co-phase
adjustment process.

Co-focus adjustment experiments were conducted on the 800mm ring segmented
mirror system using the proposed method, which involved the cross-calibration
of SHWFS with LVDT. The LVDT demonstrates a measurement accuracy better than 3
$\mu$mRMS. Compared to the SHWFS detection system, the proposed fine co-focus
adjustment method(when using 10 measurement points) improves the co-focus
adjustment accuracy of the experimental system from 91$\mu$m(PTV) to
26$\mu$m(PTV). The residual pistons for all segments fall within the capture
range of the broadband PSF phasing algorithm($\pm$15 $\mu$m). Moreover, the
LVDT has a "0 point" that can be used to record the segment’s state during the
co-focusing, which is beneficial for the segment to return to the co-focus
state quickly. This fine co-focus adjustment method, employing LVDT and SHWFS
cross-calibration, is also a suitable reference scheme for CGST’s fine co-
focus adjustment. Furthermore, the "0 point" of the LVDT may experience drift
due to environmental conditions. Therefore, further experimental research is
necessary to address this potential issue.

Funding National Natural Science Foundation of China (12273109);Yunnan
Province Young and Middle-aged Academic and Technical Leaders Reserve Talents
Project (202405AC350004);Yunnan Key Laboratory of Solar Physics and Space
Science(202205AG070009);Yunnan Revitalization Talent Support
Program(202305AS350029, 202305AT350005);Yunnan Provincial Science and
Technology Department(202103AD150013).

Acknowledgments We would like to express our heartfelt gratitude to the
Laboratory of Astronomical Technologies of Yunnan Observatories, Chinese
Academy of Sciences, for its strong support and assistance throughout this
research. The laboratory’s facilities and resources have been instrumental in
ensuring the smooth progress of our study.

Disclosures The authors declare that there are no conflicts of interest
related to this paper.

Data availability Data underlying the results presented in this paper may be
obtained from the authors upon reasonable request.

## References

  * [1] Z. Liu, Y. Deng, and H. Ji, “The chinese giant solar telescope,” Proceedings of the International Astronomical Union 8, 349–354 (2013).
  * [2] Z. Liu, H. Ji, Z. Jin, _et al._ , “Science cases in the integrated modeling of Chinese Giant Solar Telescope,” in _Integrated Modeling of Complex Optomechanical Systems II,_ vol. 10012 M. Riva, ed., International Society for Optics and Photonics (SPIE, 2016), p. 1001204.
  * [3] Z. Liu, Y. Deng, Z. yu Jin, and H. Ji, “Introduction to the chinese giant solar telescope,” in _Other Conferences,_ (2012).
  * [4] G. Chanan, M. Troy, F. Dekens, _et al._ , “Phasing the mirror segments of the keck telescopes: the broadbandphasing algorithm,” Appl. Opt. 37, 140–155 (1998).
  * [5] J. Swiegers and H. Gajjar, “Completion of the Southern African Large Telescope (SALT) primary mirror system,” in _Ground-based Telescopes,_ vol. 5489 J. M. O. Jr., ed., International Society for Optics and Photonics (SPIE, 2004), pp. 881 – 891.
  * [6] D.-Q. Su and X.-Q. Cui, “Active optics in lamost,” Chinese Journal of Astronomy and Astrophysics 4, 1 (2004).
  * [7] M. J. Wolf, M. H. Ward, J. R. A. Booth, and B. Roman, “Polarization shearing laser interferometer for aligning segmented telescope mirrors,” in _SPIE Astronomical Telescopes + Instrumentation,_ (2003).
  * [8] A. Wirth, T. Gonsiorowski, J. E. Roberts, _et al._ , “Developing and testing an optical alignment system for salt’s segmented primary mirror,” in _SPIE Astronomical Telescopes + Instrumentation,_ (2004).
  * [9] Y. Zhang, “Progress of LAMOST wavefront sensing,” in _Ground-based and Airborne Telescopes II,_ vol. 7012 L. M. Stepp and R. Gilmozzi, eds., International Society for Optics and Photonics (SPIE, 2008), p. 70123H.
  * [10] A. Ardeberg and T. Andersen, “Matching and improving the best atmospheric turbulence conditions with very large telescopes,” New Astronomy Reviews 42, 557–562 (1998).
  * [11] M. Bonamente, _Systematic Errors and Intrinsic Scatter_ (Springer Nature Singapore, Singapore, 2022), pp. 315–322.
  * [12] M. Schoeck and G. Chanan, “Analysis of edge jumps in multi-wavelength phasing of segmented-mirror telescopes,” Appl. Opt. 62, 6760–6770 (2023).
  * [13] G. Chanan, M. Schoeck, and M. Troy, “Approach based on $\chi$2 to phasing segmented-mirror telescopesusing multiple wavelengths: data reduction, wavelength selection, capturerange,” Appl. Opt. 61, 935–944 (2022).
  * [14] N. A. Roddier, “Atmospheric wavefront simulation and zernike polynomials,” in _Astronomical Telescopes and Instrumentation,_ (1990).
  * [15] X. Ke and J. Wu, _Adaptive Optics Correction_ (Springer Nature Singapore, Singapore, 2022), pp. 293–340.
  * [16] J. W. Hardy and L. A. Thompson, “Adaptive optics for astronomical telescopes,” Physics Today 53, 94–94 (1998).
  * [17] S. Yuan, Z. Jin, Y. Li, and Z. Liu, “Effect of atmospheric turbulence on optical tip measurement of active control of ring segmented telescope,” ACTA OPTICA SINICA 32 (2012).
  * [18] B. L. McGlamery, “Restoration of turbulence-degraded images$\ast$,” J. Opt. Soc. Am. 57, 293–297 (1967).
  * [19] C. Innocenti and A. Consortini, “Refractive index gradient of the atmosphere at near ground levels,” Journal of Modern Optics 52, 671–689 (2005).