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Experimental validation of hybrid optical-digital imaging
system for extended depth-of-eld based on co-
optimized binary phase masks
Alice Fontbonne, Hervé Sauer, Caroline Kulcsár, Anne-Lise Coutrot, François
Goudail
To cite this version:
Alice Fontbonne, Hervé Sauer, Caroline Kulcsár, Anne-Lise Coutrot, François Goudail. Experimental
validation of hybrid optical-digital imaging system for extended depth-of-eld based on co- optimized
binary phase masks. Optical Engineering, SPIE, 2019, 58 (11), pp.1. �10.1117/1.OE.58.11.113107�.
�hal-02453831�
Experimental validation of hybrid optical-digital imaging system for
extended depth of field based on co-optimized binary phase masks.
Alice Fontbonne
*,1
, Herv
´
e Sauer
1
, Caroline Kulcs
´
ar
1
, Anne-Lise Coutrot
1
, Franc¸ois
Goudail
1
1
Universit
´
e Paris-Saclay, CNRS, Institut d’Optique Graduate School, Laboratoire Charles Fabry, Palaiseau Cedex,
France
Abstract. We experimentally investigate the performance of co-optimized hybrid optical-digital imaging systems
based on binary phase masks and digital deconvolution for extended depth of field (DoF) under narrow-band illumi-
nation hypothesis. These systems are numerically optimized by assuming a simple generic imaging model. Using
images of DoF targets and real scenes, we experimentally demonstrate that in practice, they actually reach the DoF
range for which they have been optimized. Moreover, they are shown to be robust against small mask manufactur-
ing errors and residual spherical aberration in the optical system. These results demonstrate that the optical/digital
optimization protocol based on generic imaging model can be safely used to design DoF-enhanced imaging systems
aimed at real-world applications.
Keywords: image processing, binary phase masks, deconvolution, optical system codesign, wavefront coding, exper-
imental validation.
*Corresponding author, alice.fontbonne@institutoptique.fr
This pre-print has been published as: Alice Fontbonne, Herv
´
e Sauer, Caroline Kulcs
´
ar, Anne-Lise Coutrot, Franc¸ois
Goudail, “Experimental validation of hybrid optical-digital imaging system for extended depth-of-field based on co-
optimized binary phase masks.,” Opt. Eng. 58(11), 113107 (2019)
DOI: https://doi.org/10.1117/1.OE.58.11.113107.
1 Introduction
It is possible to enhance the depth of field (DoF)
of an imaging system without reducing the light
throughput by placing a phase mask in its aper-
ture stop. Since these masks tend to blur im-
ages, digital deconvolution has to be performed
on the acquired raw images to obtain a good
visual quality. This joint use of a phase mask
and of digital deconvolution constitutes “hybrid
optical-digital” imaging system. This concept
has been introduced by Cathey and Dowski
1
for
continuous phase DoF enhancing masks, gener-
alized by Robinson and Stork
2
to other optical
design tasks, and applied to the optimization of
DoF enhancing phase masks for different imag-
ing applications
3–14
.
Among the many possible phase mask struc-
tures, binary annular phase masks have the ad-
vantage of being easier to manufacture than non-
rotationally symmetric masks, such as cubic
phase masks.
15
They have been implemented
in practice and have shown good performance
on specific applications in infrared
16, 17
and vis-
ible imaging systems
18–21
. A systematic study
of their theoretical performance and of their ro-
bustness to aberrations has been done in a pre-
vious work
11
where we addressed in particular
the fundamental questions of the maximum DoF
reachable with these masks and of the minimum
number of rings necessary to reach a given level
of performance. It was shown that nearly op-
timal performance can be obtained with a lim-
ited number of rings and that imaging perfor-
mance is robust to small amounts of aberrations.
1
This study was based on numerical optimiza-
tions assuming a generic scene model, a given
noise level and theoretical models of spherical
aberrations. However, there remain a number of
questions about these results : how do the masks
optimized in this way perform in practice, since
observed real-world scenes may not be consis-
tent with the generic image model used for their
optimization ? In particular, do these masks re-
ally achieve the expected DoF extension? Are
they robust to aberrations and manufacturing de-
fects? The purpose of the present article is to an-
swer these questions. For that purpose, we man-
ufacture three generic phase masks optimized
for three different DoF ranges and perform a
systematic and quantitative experimental valida-
tion of an actual hybrid optical-digital imaging
system based on these masks. The obtained re-
sults are shown to validate the numerical op-
timization protocol based on generic imaging
model: this protocol can thus be safely used
to improve DoF in real-world imaging applica-
tions.
This paper is organized as follows. In Sec-
tion 2 we briefly review the basics of binary
phase mask optimization using a generic imag-
ing model. In Section 3, we describe the optical
setup used for experiments and how the masks
have been manufactured and tested. In Sec-
tion 4, we establish the optical performance of
the imaging systems including the manufactured
masks by measuring their Modulation Transfer
Functions (MTF) and their through-focus trans-
fer functions. We also characterize the DoF ex-
tension ability of the global optical-digital sys-
tem by analyzing its effective-MTFs at different
defocus values. In Section 5, we evaluate the
DoF extension provided by these masks on two
types of real scenes: a printed circuit board and a
floral scene. We also demonstrate the robustness
of the imaging system to spherical aberration.
22
2 Phase mask optimization with a generic
imaging model
Binary annular phase masks are static spatial
phase modulating optical elements consisting
of a series of N concentric annular regions of
phase modulation of alternatively 0 or π radi-
ans at the nominal wavelength λ, usually cho-
sen in the middle of the working spectral range
(see Fig. 1). Each annular constant-phase area
corresponds to a so-called ring, so that an N-
ring mask of clear aperture radius R is parame-
terized by N − 1 free normalized radius values
φ = {ρ
1
, ..., ρ
N−1
}, where the radius of the i’th
phase transition is r
i
= ρ
i
R and φ satisfies the
conditions 0 < ρ
1
< ... < ρ
N−1
< ρ
N
= 1.
Such masks can be used for extending the
DoF of an imaging system. For that purpose, the
mask is placed in the aperture stop of the imag-
ing lens. Let us assume that one observes an ob-
ject at a certain distance z
o
of the imaging sys-
tem. For that purpose, we use an optical system
of effective focal length f together with a sensor
located at a fixed imaging distance z
i
. A well-
focused object is positioned at a distance z
o0
of
the imaging system and a defocused object is
positioned at the object distance z
o
= z
o0
+ ∆z
o
of the imaging system. The defocus parameter
is defined as:
ψ(∆z
o
) =
(z
i
× NA)
2
2
1
z
i
+
1
z
o0
+ ∆z
o
−
1
f
(1)
where NA is the image Numerical Aperture of
the system. It gives the peak-to-valley optical
path difference of the Seidel defocus aberration
that takes place for such an axial object shift
from the nominal focus. Note that the defocus
parameter ψ is equal to zero when ∆z
o
= 0. If
one moves the object, the optical system is de-
focused. The induced object defocus distance
∆z
o
is negative when the object gets closer to
the optical system, and positive when the object
gets farther from the optical system. The behav-
ior of a hybrid system containing an optimized
binary phase mask with a targeted defocusing
2
(a) (b) (c)
Fig 1 The three chosen masks. (a) the “1λ” phase mask (b) the “2λ” phase mask and (c) the “2.5λ” phase mask. A
ring is defined as an annular region with constant phase modulation. Dark gray areas induce a phase shift of 0 and light
gray areas induce a phase shift of π radians at the nominal wavelength λ. (The “1λ”, “2λ” and “2.5λ” mask names
are their targeted defocus ranges as explained in the text below)
range of [0, ψ
max
] should be different depending
on whether |ψ| < |ψ
max
| or |ψ| > |ψ
max
|.
The image produced on the sensor can be
modeled by h
φ
ψ
(r) ∗ O(r), where O(r) is the
ideal sampled scene image (r represents the spa-
tial coordinates), ∗ denotes the convolution op-
erator, and h
φ
ψ
(r) is the Point Spread Function
(PSF) of the optical system. In this work, we
assume that the sampling satisfies the Shannon-
Nyquist condition. The PSF depends on the de-
focus parameter ψ related to the object distance
and on the phase function of the mask repre-
sented by the parameter set φ. We assume that
the lens is otherwise ideal (with no-aberration,
except defocus and phase mask behavior) with,
thus, no other free parameter. This acquired im-
age is then deconvolved with a digital deconvo-
lution filter w(r) to restore its sharpness. The re-
stored image at the output of this hybrid optical-
digital system can be modeled as:
ˆ
O(r) = w(r) ∗
h
h
φ
ψ
(r) ∗ O(r) + n(r)
i
(2)
where n(r) is the detection noise. The recon-
struction mean-squared error (MSE) is then de-
fined as:
MSE(φ, ψ) = E
Z
ˆ
O(r) − O(r)
2
dr
(3)
where E[·] represents the mathematical expec-
tation over the noise n(r) and the scene image
O(r), which are both assumed to be zero-mean,
stationary random processes of power spectral
density (PSD) S
nn
(ν) and S
oo
(ν) respectively,
with ν representing the spatial frequency coor-
dinates. The co-design goal is to simultane-
ously find the deconvolution filter w(r) and the
mask parameters φ that minimize MSE(φ, ψ)
over a defocus parameter set such that ψ ∈
ψ
1
, ψ
2
, · · · , ψ
K
, K being the number of defocus
parameter values. In the following, the maxi-
mal value of the defocus parameter is denoted
by ψ
max
with 0 ≤ |ψ
k
| ≤ |ψ
K
| = |ψ
max
|: it
corresponds to the limit of the targeted defocus
range of the system. For deconvolution, we use
the averaged Wiener filter that is known to min-
imize
P
k
MSE(φ, ψ
k
) and has the following
expression:
4
ew(ν) =
1
K
P
K
k=1
[
e
h
φ
ψ
k
(ν)]
?
1
K
P
K
k=1
e
h
φ
ψ
k
(ν)
2
+
S
nn
(ν)
S
oo
(ν)
(4)
where
?
stands for complex conjugate and efor
the Fourier transform. It is important to point
out that, for a given phase mask, a unique fil-
ter is used for deconvolution over the whole
field of view, regardless of the object defocus-
ing. The masks designed in Ref.
11
and man-
ufactured for the present study have been opti-
mized using a generic ideal image model with
3
a power-law PSD
23, 24
S
oo
(ν) = Kν
−a
, the pa-
rameter a being fixed to 2.5. The white noise
PSD is such that the signal-to-noise ratio (SNR)
on the raw image is 34 dB, with SNR =
10 log
10
R
S
oo
(ν)dν/
R
S
nn
(ν)dν
. With this
choice of deconvolution filter, the parameters
φ
opt
of the optimal mask are optimized using a
“minmax” criterion:
φ
opt
= arg min
φ
n
max
k
[MSE(φ, ψ
k
)]
o
. (5)
The variation of image quality as a function of
the number N of the mask rings was studied
in Ref.
11
with a numerical global optimization
method. It was shown that for a targeted defocus
range of ψ
max
= 1λ, three rings are sufficient
to reach the maximal performance, whereas
six rings are preferable for ψ
max
= 2λ and
seven rings for ψ
max
= 2.5λ. Optimal masks
for these three DoF ranges are represented in
Fig. 1 and the normalized radii of their rings
are given in Tab. 1. In the following, they will
be denoted, respectively, “1λ-phase mask”, “2λ-
phase mask”, and “2.5λ-phase mask”. These
masks have been optimized using the simple
generic and aberration-less imaging model de-
scribed above. Our goal in this article is to ex-
perimentally validate their DoF extension per-
formance on real-world scenes and in the pres-
ence of manufacturing defects and aberrations in
the optical system. For that purpose, they have
been manufactured and included in a specifi-
cally designed imaging system.
3 Experimental setup
In this section, we describe the masks manufac-
turing together with the verification of their op-
tical quality. We then describe the optical setup
we designed to evaluate their DoF enhancing
performance.
3.1 Mask manufacturing
The binary phase masks have been manu-
factured on quartz substrates using UV pho-
tolithography associated with Ion Beam Etching
(IBE) followed by Inductive Coupled Plasma
(RIE - ICP) etching. As they are used with a
narrow spectral range illumination centered on
the nominal wavelength λ, the etching depth h
is calculated so that the phase shift between two
rings is equal to π, according to the formula:
h =
λ
2(n − 1)
(6)
where n is the refractive index of quartz at
λ = 625 nm. Since quartz is a slightly bire-
fringent material and as we employ unpolarized
illumination, we use an empirical average in-
dex n =
3
4
n
o
+
1
4
n
e
where n
o
and n
e
are, respec-
tively, the ordinary and extraordinary refractive
indices. This leads to a required etching depth
of h = 573 nm, on which we allow for a man-
ufacturing tolerance of ±20 nm. This tolerance
corresponds to 3% of the total depth and does
not lead to any significant modification of the
simulated MTF, while being reasonable regard-
ing the manufacturing process.
The manufactured masks have been verified
with a 1D mechanical profilometer to check the
etching depth. An example of results obtained
with the 2.5λ-mask is given in Fig. 2. It shows
that this mask fits well with the manufacturing
specifications. It is also the case of the 1λ-
mask, whose step height has also been measured
at about 573 nm. The manufactured 2λ-mask
deviates a little more from specifications: the
height of the three steps are 559 nm for the larger
ring, 566 nm and 550 nm for the smaller ones.
Simulations performed with the CodeV
R
soft-
ware show that this deviation from the ideal step
height induces only slight changes of the MTF
with somewhat asymmetric behavior in the DoF
range relatively to the nominal focus behavior.
3.2 Optical system
In Ref.
11
, the masks are optimized assuming
that they are located exactly at the pupil of the
image-forming optics. It is also assumed that the
4
