Analysis of Strehl ratio limit with superresolution
binary phase filters
Vidal F. Canales*, Pedro J. Valle, and Manuel P. Cagigal
Departamento de Física Aplicada, Universidad de Cantabria, Santander 39005, Spain
*Corresponding author: fernancv@unican.es
Received February 26, 2016; accepted May 5, 2016; posted online June 13, 2016
Several pupil filtering techniques have been developed in the last few years to obtain transverse superresolution
(a narrower point spread function core). Such a core decrease entails two relevant limitations: a decrease of the
peak intensity and an increase of the sidelobe intensity. Here, we calculate the Strehl ratio as a function of
the core size for the most used binary phase filters. Furthermore, we show that this relation approaches the
fundamental limit of the attainable Strehl ratio at the focal plane for any filter. Finally, we show the calculation
of the peak-to-sidelobe ratio in order to check the system viability in every application.
OCIS codes: 100.6640, 110.1220, 350.5730.
doi: 10.3788/COL201614.071101.
The term superresolution has been widely used in scien-
tific literature with two different meanings: the increase
of the spatial frequency cutoff in an optical system, on
one hand, and the increase of the generalized Rayleigh res-
olution on the other. Here, we will use the latter meaning,
introduced in the classical works by Toraldo di Francia
[1]
,
which can also be associated with a decrease of the point
spread function (PSF) core width. Such a decrease can be
useful in many applications
[2–6]
, and has been analyzed in
many works
[7–9]
. The first attempts of PSF engineering
were related to amplitude filters, but in recent years
phase and hybrid designs
[9,10]
(including ternary optical
elements
[11,12]
) have been developed due to their better
performance in most applications
[13,14]
. Among all filters,
binary 0-π phase filters
[15]
stand out because they do not
produce focus displacement, achieve excellent perfor-
mances, allow several analytical results, and are easily
fabricated. For these reasons they have been proposed for
several applications
[16–19]
. Moreover, as will be seen, these
filters provide the highest PSF peak for a certain resolu-
tion without focus displacement.
Toraldo di Francia already realized that theoretically
an optical instrument with a given pupil size can yield
as a narrow a PSF core as desired, but with the unavoid-
able cost of energy loss. The quantification of such energy
loss is of utmost relevance because it can determine the
viability of a technique in an application. In the crucial
case of binary 0-π filters, this limit has been estimated
analytically
[15]
by using parabolic approximation. Such
approximation is not accurate enough, so the main goal
of this work is t o obtain a more accurate calculation of
energy loss for such filters, which are the ones used most
because of their known excellent performance.
Moreover, we will demonstrate that, for a given peak
height, binary two-zone 0-π phase filters yield the best res-
olution at the focal plane of all real filters. Hence, this
relation between the energy loss and the resolution can
be considered as a fundamental limit for such a filter’s
performance (at this point we must note that we are con-
sidering systems with rotational symmetry). This limit is
relevant for practical and theoretical reasons. The best
attempts to derive a fundamental limit was performed by
Sales and Morris
[20,21]
, who derived an upper bound for the
Strehl ratio limit from an expansion of the Bessel function
of order zero in the expression of the field at the focal
plane. This theoretical bound works very well for low
Strehl values, though it overestimates the limit for higher
values, so Sales and Morris solved this issue by substitut-
ing a region of their curve by a linear fit based precisely on
0-π binary filters. Here, we will derive the limit in a differ-
ent way, which has results that are even a bit more restric-
tive. To this aim, we will follow this scheme. First, we will
derive the relation between resolution and energy loss by
using common figures of merit. Secondly, we will demon-
strate that 0-π filters yield the highest peak for a fixed res-
olution of all two-zone filters. Then, we will demonstrate
that two-zone filters outperform filters with a higher num-
ber of zones (which includes continuous filters). These
demonstrations show that 0-π filters outperform any other
real filter, which means that the calculated relation is a
fundamental limit for the pupil filtering techniques with
no focus displacement. Furthermore, the comparison of
0-π filte rs with complex filters is discussed. Finally, reso-
lution improvement gives rise not only to a Strehl de-
crease, but also to an increase in the sidelobe intensity
[22]
.
In most applications, the sidelobe height is as crucial as
the Strehl ratio and the resolution to make a technique
viable. Consequently, we show the calculation of the peak-
to-sidelobe ratio corresponding to the limit case.
In order to relate the resolution and the energy loss, we
need some figures of merit that describe t he PSF. We have
chosen the same figures of merit as Sales and Morris: the
Strehl ratio S, defined as the ratio of the intensity at the
focal point corresponding to an unobstructed pupil, and
the normalized spot size G, which is the ratio of the radius
of the first zero of the superresolving diffraction pattern to
COL 14(7), 071101(2016) CHINESE OPTICS LETTERS July 10, 2016
1671-7694/2016/071101(4) 071101-1 © 2016 Chinese Optics Letters
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