Hybrid approach used for extended image-based
wavefront sensor-less adaptive optics
Bing Dong (董 冰)* and Ji Yu (喻 际)
School of Optoelectronics, Beijing Institute of Technology, Beijing 100081, China
*Corresponding author: bdong@bit.edu.cn
Received December 10, 2014; accepted January 16, 2015; posted online March 13, 2015
The stochastic parallel gradient descent (SPGD) algorithm is widely used in wavefront sensor-less
adaptive optics (WSAO) systems. However, the convergence is relatively slow. Modal-based algorithms usually
provide much faster convergence than SPGD; however, the limited actuator stroke of the deformable mirror
(DM) often prohibits the sensing of higher-order modes or renders a closed-loop correction inapplicable. Based
on a comparative analysis of SPGD and the DM-modal-based algorithm, a hybrid approach involving both
algorithms is proposed for extended image-based WSAO, and is demonstrated in this experiment. The hybrid
approach can achieve similar correction results to pure SPGD, but with a dramatically decreased iteration
number.
OCIS codes: 110.1080, 110.3010.
doi: 10.3788/COL201513.041101.
In wavefront sensor-less adaptive optics (WSAO) systems,
a distinct wavefront sensor is absent and a wavefront
corrector is directly driven to optimize a metric function
related to image quality. The control algorithms in WSAO
can be divided into two categories: model-free algorithms
and modal-based algorithms. Model-free algorithms like
hill climbing
[1]
, genetics
[2]
, and stochastic parallel gradient
descent (SPGD)
[3]
have been widely used in many scenar-
ios. Their common disadvantages are that a large number
of iterations are usually needed and a global convergence is
not always guaranteed. In modal-based WSAO, the wave-
front aberration is decomposed into specific modes like
Zernike modes
[4]
, Lukosz modes
[5,6]
, or deformable mirror
(DM) modes
[7,8]
. It has been demonstrated that DM modes
are superior to analytical modes like the Lukosz modes,
since the mode fitting error can be mostly avoided, espe-
cially for DMs with a low actuator number
[7,8]
. The modes’
coefficients are calculated directly from the relationship
between the mode coefficient and a proper metric
function. Modal-based algorithms lead to a much faster
convergence than model-free algorithms, and also avoid
dropping into the local optimum.
Both model-free and modal-based algorithms can be
adapted to a point-like source or an extended target.
For high-resolution biological microscopy or earth obser-
vation purposes
[9,10]
, wavefront aberration should be
corrected from an extended image. In this Letter, the
performances of the SPGD and the DM-modal-based
algorithm used for extended image-based WSAO are
evaluated and compared by simulation. From the simula-
tion result, the advantages and drawbacks of the two
algorithms are revealed. Then, a hybrid approach involv-
ing both of them is proposed to avoid drawbacks, and is
further demonstrated by experiment.
SPGD is believed to be one of the fastest model-free
algorithms. In a SPGD, small random perturbations are
applied to all control parameters (voltages of actuators)
simultaneously. Then, the gradient variation of a metric
function is evaluated to update the search direction.
The control signals are updated according to the
following rule:
u
k
¼ u
k−1
þ γδu
k
δJ
k
; (1)
where u ¼fu
1
; u
2
; …; u
N
g is the control signal vector, N is
the number of actuators, and k is the iteration number.
Here, γ is the gain factor, δu denotes small random pertur-
bations that have identical amplitudes and Bernoulli
probability distributions, and δJ is t he variation of the
metric function.
For extended image targets, several metric functions
might be used in a SPGD
[11]
. Here, we use the normalized
image sharpness function that is immune to intensity fluc-
tuations in the light source and the sensitivity variation of
the detector
[12]
J ¼
P
I
2
ðx; yÞ
P
I ðx; yÞ
2
; (2)
where I ðx; yÞ is the intensity distribution at the image
plane.
In modal-based WSAO, the low spatial frequency con-
tent of the extended-image spectral density S
J
ðmÞ is used
as the metric function
[6,8]
gðM
1
; M
2
Þ¼
Z
2π
0
Z
M
2
M
1
S
J
ðmÞmdmdξ; (3)
where M
1
and M
2
are the normalized spatial frequencies,
ξ is the angle of the spatial frequency, and m ¼
ðm cos ξ; m sin ξÞ. The relationship between the wave-
front aberratio n Φ and the metric function g is given by
COL 13(4), 041101(2015) CHINESE OPTICS LETTERS April 10, 2015
1671-7694/2015/041101(5) 041101-1 © 2015 Chinese Optics Letters