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MNRAS 527, 7036–7046 (2024) https://doi.org/10.1093/mnras/stad3642
Advance Access publication 2023 No v ember 23
Interferometric apodization by homothety – II. Experimental validation
J. Chafi ,
1 , 2 ‹Y. El Azhari,
1 , 2 , 3 ‹O. Azagrouze,
1 , 2 , 3 A. Jabiri,
1 , 2 A. Boskri ,
1 , 2 Z. Benkhaldoun
1 , 2
and A. Habib
1 , 2 , 3
1
LPHEA, Facult
´
e des Sciences Semlalia, Universit
´
e Cadi Ayyad, Av. Prince My Abdellah, BP 2390 Marr akec h, Morocco
2
Oukaimeden Observatory, Cadi Ayyad University, 40273 Marr akec h, Morocco
3
Centre R
´
egional des M
´
etiers de l’Education et de la Formation de Marr akec h, 40000 Marr akec h, Morocco
Accepted 2023 No v ember 22. Received 2023 October 25; in original form 2023 August 14
A B S T R A C T
This work presents the results of experimental laboratory tests on the apodization of circular and rectangular apertures using
the Interferometric Apodization by Homothety (IAH) technique. The IAH approach involves splitting the amplitude of the
instrumental PSF into two equal parts. One of the two produced PSFs undergoes homothety to change its transverse dimensions
while its amplitude is properly controlled. The two PSFs are then combined to produce an apodized image. The diffraction wings
of the resulting PSF are subsequently reduced by some variable reduction factor, depending on an amplitude parameter γand
a spread parameter η. This apodization approach was implemented in the laboratory using an interferometric set-up based on
the Mach–Zehnder Interferometer (MZI). The experimental results exhibit a strong agreement between theory and experiment.
For instance, the average experimental contrast obtained at a low angular separation of 2.4 λ/ D does not exceed 5 ×10
−4
. This
work also allowed us to study the influence on the apodizer’s performance of some parameters, such as the wavelength and the
density of the neutral filters.
Key words: instrumentation: high angular resolution – instrumentation: interferometers – techniques: high angular resolution –
techniques: interferometric.
1 INTRODUCTION
In high-dynamic range imaging, apodization is a commonly used
technique for enhancing the performance of coronagraphs (Soummer
2005 ; Soummer et al. 2009 ). In the field of direct detection of
exoplanets, the utility of apodized pupils has been studied by
Nisenson & Papaliolios ( 2001 ). Aime, Soummer & Ferrari ( 2001 )
suggested using interferometry for pupil apodization, and later,
Aime, Soummer & Ferrari ( 2002 ); Soummer, Aime & Falloon ( 2003 )
showed that the rejection produced by a Lyot coronagraph could be
impro v ed by apodizing the entrance pupil with a prolate spheroidal
function.
Subsequently, Soummer ( 2005 ) demonstrated that for a random
pupil, solutions based on prolate functions could also checked the
performance of the Lyot coronagraph. Aime ( 2005 ) proposed apodiz-
ing a circular aperture using an MZI and employing thin lenses with
opposite vergence to achieve apodization. This apodization technique
was experimentally verified by El Azhari et al. ( 2005 ) and Azagrouze,
El Azhari & Habib ( 2008 ) using a Michelson interferometer (MI),
and also by Carlotti et al. ( 2008 ) using a MZI. Furthermore, N’diaye
et al. ( 2012 ) explored the application of an apodization concept,
the ‘Four Quadrant Zeroth Order Grating (4QZOG)’, developed
by Mawet et al. ( 2005 ), in coronagraphy. They demonstrated that
using coloured apodization applied before a ‘Dual Zone Phase Mask
E-mail: chafi.jamal2@gmail.com (JC); youssef.elazhari@men.gov.ma
(YEA)
(DZPM)’ coronagraph can enhance the contrast performance o v er
a broader spectral band, thereby optimizing performance in the
presence of various types of noise.
The grey-scale apodizers were converted into binary masks for
manufacturing reasons, leading to the emergence of Apodized Pupil
Lyot Coronagraphs (APLC; N’Diaye et al. 2016 ; Zimmerman et al.
2016 ), used in instruments such as VLT/SPHERE (Guerri et al. 2011 )
and Gemini/GPI (Si v aramakrishnan et al. 2010 ). These coronagraphs
are also planned for future space telescopes such as the Roman
Space Telescope ( RST ; Zimmerman et al. 2016 ), LUVOIR-A, and
LUVOIR-B (Stahl, Shaklan & Stahl 2015 ; Leboulleux et al. 2017 ;
Stahl 2017 ; Leboulleux et al. 2018a ; Leboulleux et al. 2018b ;
Laginja et al. 2019 ; The LUVOIR Team 2019 ; Laginja et al. 2020 ;
Stahl, Nemati & Stahl 2020 ; Laginja et al. 2021 ; Juanola-Parramon
et al. 2022 ). Second-generation ground-based imagers, such as the
Planetary Camera and Spectrograph (PCS) for the ELT (Kasper,
Verinaud & Mawet 2013 ), the Planetary Systems Imager (PSI)
for the Thirty Meter Telescope (TMT; Fitzgerald et al. 2019 ) and
GMagAO-X for the Giant Magellan Telescope (GMT; Males et al.
2018 ) will target fainter and closer planets to their host stars, ranging
from young Jupiter’s to terrestrial planets. The Astro2020 Decadal
Surv e y by NASEM (National Academies of Sciences, Engineering,
and Medicine 2021 ) also supports the development of large space
telescopes to observe Earth-like exoplanets, such as LUVOIR-B (The
LUVOIR Team 2019 ) and Habitable Exoplanet (HabEx; Gaudi et al.
2020 ).
Carlotti ( 2013 ) and Carlotti, Pueyo & Mawet ( 2014 ) focused their
research on the apodization of vortex phase mask coronagraphs for
© 2023 The Author(s).
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License ( https:// creativecommons.org/ licenses/ by/ 4.0/ ), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited.
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Interferometric apodization by homothety 7037
MNRAS 527, 7036–7046 (2024)
arbitrary apertures, especially axially obstructed circular apertures
(including spiders). They stated that apodizers could be optimized
for the application of phase masks on these apertures and compared
these results to those of E-VLT type apertures.
Habib et al. ( 2010 ) and Azagrouze et al. ( 2010 ) introduced a
new approach to interferometric apodization, called IAH technique,
which utilizes homothety. This method yields results similar to
those obtained with a circular aperture using step-like transmission.
We refer to this transmission as being obtained using two nested
circular symmetric step functions with respective aperture radii R
1
and R
2 < R
1
. The IAH technique efficiently achieves very high
dynamic imaging by minimizing diffraction noise in the final images.
Specifically, 93.6 per cent of the stellar light is concentrated within
the central diffraction spot. This could make it a valuable complement
to commonly used coronagraphs in major observatories, such as
APLC, Vo r t e x , and others, enhancing their performance.
The IAH technique is based on duplicating the original PSF to
generate two identical PSFs via amplitude division. These PSFs are
subsequently coherently superimposed. The core strategy is to stretch
one PSF and modulate its amplitude, ensuring that the ne gativ e lobes
of one align with the positive lobes of the other, ef fecti vely reducing
the diffraction wings. This method facilitates versatile modulation of
the apodization to suit diverse configurations and needs.
One of the many advantages of the IAH technique is its ease of
implementation, as it does not require complex equipment and can
be easily integrated into existing coronagraphs. Furthermore, this
approach can be adapted to cascade multiple stages of IAH, thereby
enhancing the rejection of secondary PSF lobes and improving coron-
agraph performance. Ho we ver, using the IAH technique in cascading
stages presents a distinct trade-off. On one hand, it enhances the
efficiency of the apodization, offering impro v ed performance, such
as the reduction of diffracted light. On the other hand, it increases
the difficulty of adjustments and leads to signal losses, which can
potentially degrade the signal-to-noise ratio. It also uses simple
achromatic optical elements such as apertures and neutral density
filters, streamlining the apodizer assembly and reducing chromatic
aberrations. The IAH technique holds great potential as a powerful
tool for obtaining High Dynamic Range images and enhancing the
performance of existing coronagraphs.
In our previous article (Chafi et al. 2023 ), we presented simulation
results for various aperture shapes, such as circular , rectangular ,
hexagonal, and apertures similar to those used in the TMT telescope.
These results demonstrated promising performance in enhancing
coronagraph capabilities by increasing contrast at low-angular sep-
arations, making the IAH technique even more attractive for use in
major observatories. In the current paper, we aim to further this work
by presenting the experimental results of IAH technique on circular
and rectangular apertures.
While simulation results are highly valuable for understanding the
theoretical aspects of our IAH method, experimental validation is
crucial to pro v e its practical feasibility . Specifically , laboratory test-
ing of the IAH technique allows us to verify simulation results and,
most importantly, assess the sensitivity of various optical element
imperfections and sources of noise. Hence, experimental validation
is essential to confirm the ef fecti veness of our technique and establish
its credibility for practical applications. The experimental results will
also enable more precise calibration of simulation models to impro v e
performance prediction under real-world conditions.
The experimental implementation of our study involves many
key challenges. First, it requires delicate and precise manipulations,
including meticulous adjustment of the MZI, as well as rigorous
control of essential components for the IAH technique, such as
circular and rectangular apertures, and neutral density filters in
the MZI arms. Accomplishing these manipulations necessitates the
use of specialized equipment, strict adherence to a protocol, and
meticulous attention to detail to minimize experimental errors. It is
important to emphasize that the inherent difficulty in this experi-
mental implementation is una v oidable due to our study’s ambitious
and innov ati ve nature. In fact, the IAH technique itself is based on a
v ery simple (non-comple x) principle. Ho we v er, in the e xperimental
implementation, we propose, we have used a MZI, the alignment of
which, especially in polychromatic light, is not easy.
We arranged this paper as follows: in Section 2 , we provide a
detailed description of the experimental set-up used to implement
the IAH technique. Next, in Section 3 , we present the experimental
results. We started by exploring the case of a rectangular aperture
before moving on to a circular one. In the same context, we took
the opportunity to examine the chromatic behaviour of the IAH
technique using various wavelengths. Finally, Section 4 summarizes
our findings and outlines our prospects for future research.
2 EXPERIMENTAL SET-UP
The choice of the experimental set-up is of paramount importance
in the realization of IAH. The objective of this section is to provide
a concise yet sufficiently detailed description of the optical set-up
used, as well as the protocol applied for image acquisition.
2.1 Optical assembly
The experimental set-up used for the IAH technique is depicted
in Fig. 1 . It incorporates a monochromatic He–Ne laser with a
wavelength of 543.5 nm, featuring a non-adjustable cavity. To assess
the chromatic behaviour of our system, we substituted this laser
with others, whose wavelengths will be detailed in Section 3.4 , each
operating at a unique wavelength. This enabled us to e v aluate the
efficacy of our device across various wavelengths, a 5 ×microscope
objective (OC) with a focal length of 25.4 mm, and a 100 mm
lens (L
0
) to produce cylindrical beams. A folding mirror (M
0
)
is used to optimize the workspace on the Smart Tab l e M-OTS-
UT2 optical table. Our experiment relies on the use of the MZI,
assembled with two semireflective beam splitters, BS
1 and BS
2
,
and two plane mirrors, M
1 and M
2 (Fig. 2 ). The MZI has two
outputs: one with in-phase beams (additive output), allowing the
observation of the apodization effect, and the other with out-of-phase
beams (subtractive output), resulting in an anti-apodization effect.
In this case, the energy is redirected towards the secondary lobes.
This duality in the MZI’s outputs serves as a crucial component
of our validation process. It not only verifies the accuracy of
the apodization but also facilitates the measurement of the total
light source intensity. This measurement is essential for precisely
determining the intensity of an exoplanet, optimizing our results,
and enhancing the characterization of exoplanets. The spatial ar-
rangement of the different elements in the MZI facilitates the
introduction of the necessary elements for the IAH technique.
The set-up is designed to operate with single circular apertures
or rectangular apertures, depending on the desired apodization
type.
To achieve circular symmetry apodization, we used two circular
apertures, P
1
and P
2
, with respective diameters φ1
and φ2
, allowing
us to control the beam diameters with a homothety ratio η=
φ2
φ1
. The
beam intensity control was accomplished using two neutral density
filters, ND
1 and ND
2
. Both apertures and filters were carefully
positioned in the two arms of an MZI, as illustrated in Fig. 1 . A
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7038 J. Chafi et al.
MNRAS 527, 7036–7046 (2024)
Figure 1. The optical set-up, based on the MZI, was designed to operate with either single circular apertures or adjustable rectangular apertures, depending
on the desired apodization type. The set-up comprises two semireflective beam splitters, BS
1
and BS
2
, along with two plane mirrors, M
1
and M
2
. Two single
circular apertures, P
1
and P
2
, with respective diameters φ1
and φ2
(or tw o rectangular apertures) were used. Tw o neutral density filters, ND
1
and ND
2
, enabled
beam intensity control. The diameter of the incident beam with intensity I
0
is denoted as φ0
. The final focal plane to observe the apodization effect on the PSF
is P
1
, while the image plane to observe the apodized pupil is P
2
.
Figure 2. Image illustrating the interferometric part of the experiment,
implementing interferometric apodization through homothety. The light
beams go into the MZI from the left and exit from the right. Two circular
or rectangular apertures, P
1
and P
2
, are accurately positioned between the
second beam splitter and the two side mirrors to control the beam diameters.
Two neutral density filters, ND
1
and ND
2
, are placed between the side mirrors
and the first beam splitter to control the intensities of the two beams. The
notations used for the mentioned components are the same as those used in
Fig. 1 .
200 mm lens (L
s
) formed the image of the apodized aperture on the
final image plane P
2
of the instrument, allowing the observation of
the apodization effect in the focal plane P
1
of L
s
. Both planes were
observ ed successiv ely by placing the CCD camera in the appropriate
plane.
2.2 Image acquisition and processing
To capture the obtained images, we employed a high-resolution
imaging camera equipped with a KAF-8300 CCD sensor. This
sensor boasts a surface area of 17.96 ×13.52 mm
2 and features
a 3326 ×2504 pixels array, resulting in o v er 8.3 me gapix els with
each pixel measuring 5.4 μm on each side. An important attribute for
acquiring apodized PSF images is the sensor’s dynamic range or bit
depth. The CCD camera used has a 16-bit dynamic range, enabling
image encoding with more than 65 000 levels of grey-scale.
To mitigate the dark current and read out noise, the camera is
equipped with an internal thermoelectric cooling system, capable of
lowering the sensor’s temperature to as low as −40
◦C relative to
the ambient temperature. The exposure time can be set between 0.1
and 60 min. Ho we ver, we consistently opted for a minimal exposure
time and adjusted the beam intensity to a v oid pixel saturation.
Subsequently, we performed e xtensiv e image processing to correct
for dark current effects using the AstroImageJ interface (Collins et al.
2017 , AIJ).
3 EXPERIMENTAL RESULTS
In this section, we present the experimental results of the IAH
technique obtained in the laboratory using the MZI-based set-up
depicted in Fig. 1 and described in Section 2 . We will sequentially
present the results obtained with a rectangular aperture and then
those obtained with a circular aperture. Additionally, we will explore
the influence of various parameters on the performance of the
IAH technique. Specifically, we will investigate the impact of the
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Interferometric apodization by homothety 7039
MNRAS 527, 7036–7046 (2024)
wavelength of the light used. This will provide insights into the
chromatic response or chromatism of the IAH technique.
3.1 Rectangular aperture
We started by experimentally studying the case of a rectangular
aperture. The advantage of this configuration lies primarily in the
availability of high-quality rectangular apertures from optical sup-
pliers, with dimensions that can be easily and accurately controlled in
a continuous manner. This simplifies the set-up process significantly,
streamlining the adjustments needed for experimentation.
The use of this geometric aperture shape provides the advantage
of reducing secondary diffraction lobes along both ( x and y ) axes
and, more importantly, along the diagonals, as demonstrated by
Zanoni & Hill ( 1965 ). Nisenson & Papaliolios ( 2001 ) reproduced
an apodized rectangular aperture using transmission functions that
create extended regions around the PSF diagonals where the inten-
sity decreases rapidly. More recently, Itoh & Matsuo ( 2022 ) have
explored the use of a rectangular aperture to achieve deep nulling.
Before presenting the experimental results of rectangular aper-
tures, let us recapitulate some steps involved in optimizing the design
of the experimental set-up. These steps have been detailed elsewhere
in a previous article (Chafi et al. 2023 ).
The IAH technique involves superimposing in the instrument’s
focal plane two PSFs that are coherent with each other, where one is
judiciously stretched (with a stretching factor η) and its amplitude is
affected by a multiplicative factor γ. The resulting amplitude at the
MZI output in the in-phase channel is then given by:
A( u, v, γ, η) =
sin ( πu )
πu
sin ( πv)
πv
+ γsin ( πuη)
πuη
sin ( πvη)
πvη . (1)
The out-of-phase channel produces an anti-apodized amplitude. To
obtain this amplitude, we simply replace the parameter γwith −γ
in equation ( 1 ).
These amplitudes are coherent superpositions, respectively addi-
tive and subtractive, of the Fourier transforms of the transmittances
t
1
( x , y ) and t
2
( x , y ) of two rectangular apertures with respective
dimensions (
x
,
y
) and ( η
x
, η
y
), each placed in one of the two
arms of the MZI:
t
1
( x , y ) =
x
x
y
y
and
t
2
( x , y ) = γ
x
η
x
y
η
y
. (2)
Where (
x
x
) is the window function, equal to 1 for −
x
2
< x <
x
2
,
and 0 otherwise.
The ef fecti veness of the apodization greatly depends on the values
assigned to the parameters γand η. In a recent work (Chafi et al.
2023 ), we determined the optimal values for these two parameters to
concentrate the maximum energy in the central lobe of the PSF. The
optimal values are γ= 1.928 and η= 1.657. This corresponds
to concentrating 91.8 per cent of the total energy in the central
diffraction lobe, compared to 81.5 per cent without apodization.
These values constrain the useful surface areas S
1 and S
2 of the
two rectangular apertures P
1 and P
2
, as well as the densities D
1
and D
2
of the two neutral density filters ND
1
and ND
2
, through the
following two relations:
S
2
= η2
S
1 and D = D
1
−D
2
= 2 log
10
γ
η. (3)
Tab l e 1 summarizes the optimal values of parameters γand η, along
with the corresponding density difference value D . It also provides
Tab l e 1. Optimal values of the parameters γand ηfor the rectangular
geometry, along with the corresponding density difference D .
1 and
2 represent the dimensions of square apertures satisfying the optimal
apodization conditions.
γ η D
1
(mm)
2
(mm)
1.928 1.657 0.132 1 1.657
a set of possible values for the dimensions
1 and
2 of the square
apertures to be used.
To approximate the apodization parameters γand ηas closely as
possible to their optimal values γ= 1.928 and η= 1.657, we initially
used commercially available neutral density filters with respective
densities of D
1
= 0.2 and D
2
= 0.1. Additionally, we employed
adjustable rectangular apertures and set the dimension of aperture
P
1 to
1
=
1
= (1 ±0 . 05) mm (square geometry). Subsequently,
we conducted experimental adjustments to the dimension of aperture
P
2
to maximize the apodization in the P
1
plane along both transverse
directions. This was achieved with
2
=
2
= (1 . 63 ±0 . 15) mm ,
corresponding to experimental values of γexp
= 1.83 (5 per cent
deviation) and ηexp
= 1.63 (2 per cent deviation). The average overall
discrepancy between our theoretical predictions and experimental
measurements is 3.5 per cent, thereby illustrating the closeness of
our experimental results to the theoretical predictions.
Fig. 3 provides an overview of the pupil plane (column a) and the
focal plane (column b) for the cases of an unapodized pupil (top),
an apodized pupil (middle), and an anti-apodized pupil (bottom).
These images give a qualitative appreciation of the effect of IAH on
the output in-phase (or apodized output) and output in opposition
to the phase (or anti-apodized output) of the MZI, despite the
slight differences between the experimental values of the γand η
parameters of the IAH set-up and the theoretical optimal values.
In order to quantify the apodization effect, we performed cross-
sections along specific directions of the pupil plane image (Fig. 3 a)
and the focal plane image (Fig. 3 b).
Fig. 4 , on the other hand, provides intensity cross-sections in
the pupil plane displayed on a linear scale in blue compared with
the theoretical curve in red dashed lines. It is noteworthy that
there is good agreement between the theoretical prediction and the
experimental results. Indeed, the results predicted by Azagrouze
( 2012 ) are confirmed, stating that the IAH is equi v alent to a stepwise
transmission.
Fig. 5 shows the intensity cross-sections of the PSFs obtained
in the focal plane along the x -axis. The experimental curves are
displayed in blue, while the theoretical ones are shown in red.
Overall, for both the unapodized case (a) and the apodized (b)
and anti-apodized (c) cases, there is a good agreement between the
experimental results and the theoretical predictions. The apodized
PSF exhibits a slight broadening of the central lobe compared to
the unapodized PSF. Conversely, a slight narrowing of the central
lobe is observed for the anti-apodized PSF. The attenuation of
secondary diffraction lobes, theoretically predicted (Chafi et al. 2023 )
is confirmed experimentally. This translates to an average contrast
of 10
−3 at a distance between 3 λ/D and 4 λ/D. These results are
comparable to those obtained by other apodization techniques, such
as the one studied by El Azhari et al. ( 2005 ) and Azagrouze, El
Azhari & Habib ( 2008 ), who experimentally validated the interfero-
metric apodization of a one-dimensional rectangular aperture using a
Michelson interferometer. Similarly, Carlotti et al. ( 2008 ) performed
apodization of a rectangular aperture using the MZI.
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7040 J. Chafi et al.
MNRAS 527, 7036–7046 (2024)
(a) (b)
Figure 3. Rectangular apodization: column (a) represents the pupil plane,
while the column (b) displays the focal plane. At the top, we observe the
case of an unapodized pupil; in the middle, an apodized pupil; and at the
bottom, an anti-apodized pupil. The focal plane intensities are presented on
a logarithmic scale, and the corresponding theoretical expressions are given
by equation ( 1 ) for the following three cases: the unapodized case ( γ= 0),
the apodized case ( γpositive), and the anti-apodized case ( γne gativ e).
Furthermore, the rectangular geometry exhibits a very interesting
property, resulting in significant attenuation along the diagonals
y = ±x . As early as in 2001, Nisenson & Papaliolios ( 2001 )
interpreted this property as a double apodization along both the x
and y axes. Fig. 6 , we present a zoom on the diagonal regions of the
experimental results obtained with the interferometric apodization
using a rectangular aperture. The left panel displays the case without
apodization, the apodized case in the centre, and the anti-apodized
case on the right.
To validate the apodization effect, we performed normalized cuts
with the same maximum intensity along the diagonal regions on a
logarithmic scale. These cuts are shown in Fig. 7 , with the experimen-
tal images in blue and the theoretical predictions in red dashed lines.
The experimental results show a contrast of approximately 10
−3.5
for angular separation between 2 λ/D and 3 λ/D, as well as a similar
contrast for more distant separations of about 6 λ/D. Once again, a
good agreement between theory and experiment can be observed.
The IAH technique applied to rectangular geometries offers a
significant advantage o v er other apodization techniques, such as
the one proposed by Carlotti et al. ( 2008 ). Our method allows
for simultaneous apodization along both x and y ax es, pro viding
a two-dimensional (2D) solution without the necessity for further
adjustments or modifications. In contrast, the method of Carlotti et al.
( 2008 ) achieves 2D circular apodization through the incorporation
of complementary phase masks in both arms of the MZI. Yet , for
rectangular apodization, their method is unidimensional, focusing
e xclusiv ely on the x -axis. This implies that a secondary apodization
stage would be requisite to encompass both x and y axes, which
could introduce additional complexity to the system. On the other
hand, our method requires only a single apodization stage, making it
easier to implement and achieve similar or even superior performance
compared to more complex apodization methods. Additionally,
significant attenuation is observed in the diagonal regions, indicating
that the IAH technique with a rectangular geometry offers a general
and precise idea of the possibility of applying the IAH technique
to any geometric shape of a telescope. For example, segmented
apertures are similar to those used in large observatories like the
TMT telescope that we used in the simulations shown in our previous
article (Chafi et al. 2023 ).
3.2 Cir cular apertur e
In the study conducted by El Azhari et al. ( 2005 ), a Michelson inter-
ferometer was used to experiment with one-dimensional apodization
of a rectangular aperture. Subsequently, Azagrouze, El Azhari &
Habib ( 2008 ) investigated two-dimensional apodization. Carlotti
et al. ( 2008 ) performed an experimental validation of this apodization
in the laboratory using a MZI, with converging and diverging lenses
placed in the arms of the interferometer.
Following the mathematical formalism presented in our previous
article (Chafi et al. 2023 ), we have extensively examined the theory
associated with circular apertures, both unapodized and apodized,
in the image plane. As a reminder, we introduced mathematical
expressions for the transmission using a circularly symmetric top-hat
function
r
R
, where represents the circularly symmetric top-
hat function with a radius R . This function value 1 inside the circle
of radius R and 0 outside. Furthermore, the amplitude associated
with these apertures is given by 2
J
1
( πr)
πr
(Born & Wolf 1980 ), and the
apodization parameters γand ηare used to control the diameters and
amplitudes of the beams, respectively. The transmittance is given by
γ
r
ηR
, and its associated amplitude is 2 γJ
1
( ηπ r )
ηπr
. The analytical
expression of the amplitude associated with the apodized pupil in the
focal plane P
1
of the lens L
s
is expressed as follows:
( r, γ, η) = 2
J
1
( πr)
πr
+ 2 γJ
1
( πr η)
πr η, (4)
where J
1
is the first-order Bessel function.
The amplitude associated with the output in the opposite phase,
displaying the anti-apodized case, in the final focal plane P
1
, can be
expressed by equation ( 4 ) by replacing γwith −γ.
The real parameters γand ηare used to control the diameters and
amplitudes of the beams, respectively. This optimization facilitates
the superposition of the two PSFs. As a result, the pedestals of the
PSFs can be ef fecti vely compensated, leading to the achievement of
apodization.
Before implementing the IAH technique, we conducted a siz-
ing study to adapt the dimensions of the components (diffraction
apertures and neutral density filters) to the available components
in the market. This study was necessary to achieve the optimal
performance of IAH in our experiment. As described in Article
1 (Chafi et al. 2023 ), we carried out an optimization of various
parameters, including the diameters of the diffraction apertures and
the densities of the neutral density filters. These optimizations were
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Figure 4. Transverse cross-sections along the horizontal axis, expressed in millimeters, of results in the pupil plane P
2
for a rectangular aperture, corresponding
to Fig. 3 (a), are presented in blue and compared to theoretical predictions represented by red dashed curves. The positions on the graph correspond to the
non-apodized (a), apodized (b), and anti-apodized (c) pupils.
Figure 5. Theoretical predictions and corresponding experimental results are shown for the images presented in Fig. 3 (b). The positions on the graph correspond
to the non-apodized (a), apodized (b), and anti-apodized (c) PSF. The cross-sections along the x axis, with the horizontal axis in units of λ/ D , and the vertical
axis representing the normalized intensity on a logarithmic scale.
Figure 6. Zoom on the diagonal regions of the images in Fig. 3 (b), with the same position, to display the effect of IAH technique in these zones.
performed while considering the optimal values of the parameters
γ= 0.525 and η= 0.625. Through this optimization, we achieved
an energy concentration of 93.6 per cent in the central diffraction
lobe, which represents an impro v ement compared to 84 per cent
in the case without apodization. We also provided a sizing study
to find standard component values that closely match the optimal
values for users who do not have access to exact components. These
standard values were obtained while considering practical constraints
such as component availability in the market and manufacturing
limitations.
To size our experiment, we adopted standard parameters ( γ=
0.498 and η= 0.6) as presented in Section 3.3 of our previous
article (Chafi et al. 2023 ). These values were chosen to achieve
performance close to the optimal values. As for the diameter
of the diffraction aperture P
1
, we opted for φ1
= 1000 μm, a
commonly av ailable v alue in most optical component catalogs.
Using this value and the homothety ratio ( η=
φ2
φ1
), we obtained
a diameter value for the second aperture P
2 of approximately
600 μm. For the selection of the neutral density filters, we used
specific densities to achieve a difference in density D close to
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7042 J. Chafi et al.
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Figure 7. Diagonal cross-sections along x = y from the results in the pupil plane P
1
for a rectangular aperture, corresponding to Fig. 6 , are displayed in blue
and compared to theoretical predictions on a logarithmic scale, represented by red dashed curves. The positions on the graph correspond to the non-apodized
(a), apodized (b), and anti-apodized (c) PSF. The horizontal axis is scaled in units of λ/ D , while the vertical axis represents the normalized intensity (or contrast)
on a logarithmic scale.
Figure 8. Apodization with circular symmetry. Column (a) depicts the pupil
plane on a linear scale, while column (b) displays the focal plane on a
logarithmic scale, for the cases of non-apodized (top), apodized (middle),
and anti-apodized (bottom).
the optimal value of 0.15. Exactly, we chose neutral density filters
with densities of D
1
= 0.20 and D
2
= 0.04, giving a D of
0.16.
Fig. 8 presents the experimental results at the pupil plane [column
(a)] and the focal plane [column (b)] considering three different
situations: non-apodized (top), apodized (middle), and anti-apodized
(bottom). These illustrations serve to qualitatively assess the impact
of IAH on the phase outputs (apodized outputs) and the anti-phase
outputs (or anti-apodized outputs) of the MZI. To obtain a non-
apodized image, the light beam propagating along one of the arms
of the MZI is blocked using a movable screen.
The results show good agreement with the simulation results.
In comparison with the Airy disc, the apodized PSF exhibits an
enlarged central core and a substantial reduction in secondary rings.
Conversely, the anti-apodized PSF displays a slightly narrower
central core, but with higher diffraction wings, particularly in the
first two rings.
The radial profiles of the images and PSFs from Fig. 8 are shown
in Figs 9 and 10 . The experimental data are depicted in blue, while
the theoretical projections are shown in red. In all cases, whether it
is the non-apodized situation (a), apodized (b), or anti-apodized (c),
can see a significant correspondence between the experimental data
and the theoretical projections. Linear scales are used for the pupil
images and logarithmic scales for the PSFs.
To impro v e the representation, all e xperimental curv es are ob-
tained from azimuthal averaging. This method highlights the o v erall
trends of intensity profiles, making it particularly suitable for systems
with radial symmetry. The principle of azimuthal averaging involves
calculating the average of intensities along concentric circles cen-
tred on the interest point, providing a smoother and more robust
representation of intensity variations.
The experimental results demonstrate a good agreement between
theory and practice up to angular distances slightly greater than
7.5 λ/ D . In addition to the expected enlargement of the central lobe
of the PSF, a significant attenuation of the secondary rings is also
observed.
In our comparative study on the IAH technique, we juxtaposed
our experimental results with those derived from other apodiza-
tion techniques. Our findings demonstrated an average contrast of
approximately 10
−2.5 at angular separations ranging from 2 λ/ D to
3 λ/ D , and 10
−4 between 6.5 λ/ D and 8 λ/ D . In comparison, other
apodization studies have reported an attenuation of around 10
−2.5
at
approximately 3 λ/ D (Carlotti et al. 2008 ). Our results align with the
objectives of similar techniques that aim to attenuate the diffracted
stellar intensity, as illustrated by Por ( 2017 ), Asmolova et al. ( 2018 ),
and Zhang et al. ( 2018 ). These approaches, including ours, also
target the optimization of performance for interferometric devices
used in large telescopes. These comparisons underscore that our
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Figure 9. Comparison of azimuthal averages of the pupil (blue curves) from Fig. 8 (a), and the theoretical predictions (dashed red curves). The positions on
the graph correspond to the non-apodized (a), apodized (b), and anti-apodized (c) pupils. The horizontal axis is measured in millimeters, while the vertical axis
represents the normalized intensity on a linear scale.
Figure 10. Alignment between theoretical predictions (dashed red) and experimental data (blue) for circular symmetry apodization. The positions of the curves
correspond to the images from Fig. 8 (b). The positions on the graph correspond to the non-apodized (a), apodized (b), and anti-apodized (c) PSF. The horizontal
axis is measured in units of λ/ D , while the vertical axis represents the normalized intensity on a logarithmic scale.
IAH approach offers commendable contrast performance, even at
minor angular separations
3.3 Effects of neutral density filters
We have modified the transmission values T of the neutral density
filters to study their effect on our IAH technique. The transmission
coefficients T
i ( i = 1, 2) of the neutral density filter are expressed
as T
i
= 10
−D i
, where D is the transmission coefficient of the neutral
density filter given by D = D
1
−D
2
= 2 log
10
(
γ
η) .
Fig. 11 illustrates the evolution of the average azimuthal intensity
in the final focal plane with respect to the transmission T of the
neutral density filters: the blue dashed curve at 97.2 per cent ( γ=
0.58) and the green curve at 44.45 per cent ( γ= 0.4) exhibit
minimal attenuation and fail to achieve maximum contrast for
angular separations. In contrast, the red curve at T = 69 . 3 per cent
( γ= 0.498), which corresponds to the optimal value of T with
parameters γ= 0.598 and η= 0.6, shows a higher attenuation
compared to the blue and green dashed curves while maintaining a
high contrast in angular separations. In essence, the optimal value
of T strikes a fa v orable balance between filter transmission and
image contrast. In this experiment, the value of T = 69 . 3 per cent
offers a compelling compromise between attenuating diffracted light
and preserving image contrast in angular separations. Therefore,
this value of T is considered the most optimal for this particular
experiment.
3.4 Chromaticism
The effect of chromatic dispersion is a significant limitation in
many optical applications, particularly in imaging and astronomical
observations. To address this issue, various solutions have been
proposed. One such solution was presented by Wynne ( 1979 ), who
studied chromatism by using an achromatic system to produce an
image with a focal ratio proportional to 1/ λ. Additionally, Aime
( 2005 ) proposed using prolate functions to make the Apodized Pupil
Lyot Coronagraph (APLC) achromatic. The MZI has also been
employed to achieve a more precise approximation of chromatic
apodization, rather than relying solely on chromatic lenses or
multiple glasses to compensate for the chromatic effect, as studied
by Carlotti et al. ( 2008 ). Another solution was proposed by Aime
et al. ( 2010 ), who introduced a new instrumental concept called
the Achromatic Rotation-shearing Coronagraph (ARC) to eliminate
chromaticity in astronomical observations. Similarly, Guerri et al.
( 2011 ) experimentally validated an achromatic apodizer in three
bands: H (1.6 μm), J (1.191 μm), and Y (1.063 μm).
While achieving achromatic apodization for all wavelengths re-
mains a challenge, we e v aluated the chromatic effect on the IAH
technique by using three helium–neon laser sources at different
wavelengths ( λ0
= 543 . 5 nm , λ1
= 594 nm , and λ2
= 632 . 8 nm ).
The average azimuthal intensity curves in the focal plane exhibit
a shift in angular separation, as shown in Fig. 12 , but with similar
contrast levels for the three wavelengths tested, indicating that the
chromatic effect on the central lobe is very weak. In other words,
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Figure 11. The azimuthal average in the final focal plane with different values of transmission T is presented in logarithmic scale: dotted blue corresponds to
T = 97 . 3 per cent , green represents T = 44 . 45 per cent , and red indicates the optimal value of T = 69 . 3 per cent .
Figure 12. The average azimuthal intensities in the focal plane are displayed
in logarithmic scale with different colours as follows: green for λ0
=
543.5 nm, orange for λ1
= 594 nm, and red dotted line for λ2
= 632.8 nm.
The effect of chromaticity on the central lobe is illustrated by the normalized
intensity in linear scale displayed at the top.
the performance of the IAH technique is insensitive to wavelength
variations. This result is consistent with the nature of IAH, which
is based on diffraction principles that affect all wavelengths in
proportion to the aperture used. Furthermore, IAH’s ability to provide
similar performance o v er a wide range of wav elengths makes it
highly advantageous for applications requiring imaging in different
spectral bands.
Although IAH is relatively insensitive to wavelength aberrations,
broad-band PSFs pose distinct challenges due to their rich spectral
diversity. Indeed, when subjecting broad-band light to IAH, the risk
of chromatic aberrations arises, potentially affecting the quality of
the apodized PSF. Ho we ver, our preliminary data, backed by the
fundamental theory of IAH, confirm that the use of achromatic
components, such as simple apertures and neutral density filters,
allows IAH to reduce sensitivity to this spectral diversity. With this
in mind, we plan to further validate our approach by experimenting
with IAH using white light in our future studies. Finally, achromatism
is a crucial property for astronomical observation instruments as it
enables high-quality imaging across a broad range of wavelengths
without the need for constant realignment of optics to compensate
for wavelength variations.
4 CONCLUSIONS
The study presented in this article highlights the potential of the
IAH technique for improving the performance of a coronagraph in
terms of contrast at very small angular separations. Our experimental
results are in good agreement with theoretical predictions, and the
IAH method has pro v en to be competitiv e with other apodization
techniques. We have also observed a significant attenuation of the
secondary rings and a noticeable broadening of the central PSF lobe.
The contrast achieved with the IAH technique is highly competitive,
reaching up to 10
−2.5
for angular separations between 2 λ/ D and 3 λ/ D ,
and up to 10
−4
for distances between 6.5 λ/ D and 8 λ/ D .
Furthermore, our IAH approach with a rectangular geometry
demonstrates a dual apodization effect in the diagonal regions as
well as the apodization effect applied along both the x and y axes.
Other techniques, as studied by Carlotti et al. ( 2008 ), only apply to
a single axis, which complicates the application of apodization to
both the x and y axes. In contrast, the IAH technique with a rectan-
gular geometry offers a general idea of the possibility of applying
this technique to telescopes with more complex geometric shapes,
such as segmented telescopes used in large ground-based or space
telescopes.
The IAH approach can be applied to any geometric shape of
the telescope and has pro v en to be very useful in improving the
performance of commonly used coronagraphs (e.g. APLC, Vo r t ex ,
FQPM, etc.) in terms of average contrast at small separations.
Although achieving achromatic apodization for all wavelengths
remains a challenge in the direct imaging of exoplanets, the chromatic
effect on the IAH technique has been studied using three Ne–He laser
sources of different wavelengths. The results have shown that it is
possible to achieve a similar b ut wa velength-shifted contrast due to
the nature of diffraction. This indicates that the IAH technique could
provide achromatic performance. We plan to further improve the
technique by using polychromatic light in the next stage. The IAH
technique shows great promise for high-contrast imaging and is com-
parable to other similar techniques, such as those using Michelson
or MZI.
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Moreo v er, it is interesting to note that the exploration of alternative
aperture shapes, such as hexagonal or segmented, could be consid-
ered for the experimental implementation of the IAH technique.
Ho we ver, a thorough optimization study would be necessary to
assess the advantages and challenges associated with these new
aperture shapes in order to determine the optimal parameters ( γ;
η). Additionally, a procedure for acquiring new equipment would
also be required to implement these alternative experimental config-
urations. Thus, the experimental realization of IAH with hexagonal
or segmented apertures could be considered an objective of future
work aimed at deepening the understanding of this technique and
expanding its potential applications. In fact, in the experimental part
of this study, we did not address either central obstruction apertures or
segmented apertures. Ho we ver, in pre vious work (Chafi et al. 2023 ),
based on simulation calculations for IAH-apodized pupil responses,
we demonstrated the significant potential of the IAH technique in
reducing dif fracti ve ef fects, both from the outer contour and the gaps
between segment spaces. In this previous study, we used the values
of optimized parameters determined for the case of a rectangular
aperture in our simulations. It might be worthwhile to determine
optimized parameters specific to other geometries. The principle
remains the same, but the calculation procedures would need to be
modified. Transitioning to experimentation will require designing
the experimental set-up before proceeding with specific adjustments
and measurements.
The IAH technique holds promising perspectives for celestial
observations or tests, especially when combined with a coron-
agraph, allowing significant contrast enhancement and impro v ed
performance. Ho we ver, it is essential to acknowledge that real-
izing such observations entails several complex challenges that
need to be addressed. A careful examination of the technical and
logistical requirements is crucial to implement this experimental
manipulation within an observational framework, especially when
using a telescope with a minimum diameter of 6 metres to gather
sufficient starlight and achieve the resolution needed for exoplanet
detection. This involves considerations such as the availability
of appropriate equipment, precise planning of experimental con-
ditions, and managing potential technical limitations. It should
be noted that all these aspects will be the subject of future
studies.
In summary, the IAH technique presented in this article is a
promising approach to enhance the contrast of a coronagraph and can
be applied to any geometric shape of a telescope. In the next stage,
we plan to further impro v e the IAH technique by using multiple
cascaded stages to increase the contrast at very short separations.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable request
to the corresponding author.
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