# Most planets might have more than 5 Myr of time to form

Susanne Pfalzner Jülich Supercomputing Center, Forschungszentrum Jülich, 52428
Jülich, Germany  Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69,
53121 Bonn, Germany  Shahrzad Dehghani  Jülich Supercomputing Center,
Forschungszentrum Jülich, 52428 Jülich, Germany  Department of Physics,
University of Cologne, Cologne, Germany  Arnaud Michel Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, ON, K7L 3N6,
Canada<EMAIL_ADDRESS>
###### Abstract

The lifetime of protoplanetary disks is a crucial parameter for planet
formation research. Observations of disk fractions in star clusters imply
median disk lifetimes of 1 – 3 Myr. This very short disk lifetime calls for
planet formation to occur extremely rapidly. We show that young, distant
clusters ($\leq$ 5 Myr, $>$ 200 pc) often dominate these types of studies.
Such clusters frequently suffer from limiting magnitudes leading to an over-
representation of high-mass stars. As high-mass stars disperse their disks
earlier, the derived disk lifetimes apply best to high-mass stars rather than
low-mass stars. Including only nearby clusters ($<$ 200 pc) minimizes the
effect of limiting magnitude. In this case, the median disk lifetime of low-
mass stars is with 5 – 10 Myr, thus much longer than often claimed. The longer
timescales provide planets ample time to form. How high-mass stars form
planets so much faster than low-mass stars is the next grand challenges.

circumstellar matter, protoplanetary disks, open clusters and associations,
planet formation

##  1 Introduction

The disks surrounding young stars provide the building material for planets.
While terrestrial mantle rocks show that the Earth took tens of Myr to form
(Halliday & Kleine, 2006), such direct formation dating is impossible for gas
giants and exoplanets. Therefore, the frequency of disks around stars of
different ages is used as a method to obtain information about the time
available for planet formation (Haisch et al., 2001). The derived median disk
lifetime is a ”make-or-break” test for planet formation theories. As stellar
ages of individual stars are intrinsically highly imprecise (e.g., Bell et
al., 2013; Richert et al., 2018), disk fractions, $f_{d}$, in young star
clusters are used instead. As clusters consist of fairly coeval stars, their
ages can be determined with higher accuracy than for individual stars.

The decline of the disk fraction with cluster age, $t$, has been shown for
many different disk indicators, such as infrared excess or accretion
signatures (Haisch et al., 2001; Hernández et al., 2007; Fedele et al., 2010;
Ribas et al., 2014; Richert et al., 2018). Exponential fits of the form
$f_{d}(t)=exp(-t/\tau)$ provide an inconsistent picture with median disc
lifetimes ranging from 1 – 3.5 Myr to 5 – 10 Myr. As at least half the stars
in the field seem to harbour planets (e.g., Winn & Fabrycky, 2015), the
shorter disk lifetimes would imply extremely rapid planet growth.

Table 1: Examples of the fraction of old and nearby clusters in disk lifetimes studies Reference | Number of clusters | Fraction of clusters with | disc lifetime
---|---|---|---
|  |  $t_{c}\geq$ 5 Myr |  $d\leq$200 pc |
Richert et al. (2018) | 69 | 0.00 | 0.00 | 1.3 – 3.5
Haisch et al. (2001) | 7 | 0.15 | 0.28 | 3.0
Hernández et al. (2008) | 18 | 0.17 | 0.22 |  $<$ 5
Fedele et al. (2010) | 10 | 0.20 | 0.33 | 2.9
Briceño et al. (2019) | 9 | 0.55 | 0.00 | 2
Ribas et al. (2014) | 22 | 0.15 | 0.54 | 2
Ribas et al. (2015) | 11 | 0.5 | 0.73 | 5.2
Michel et al. (2021) | 11 | 0.36 | 0.91 | 8.0
this work, Fig. 1 middle | 14 | 0.50 | 1.0 | 7 – 8

The uncertainties in cluster age determination (Bell et al., 2013) is a known
problem in deducing disk lifetimes from cluster disk fractions. Besides,
environmental effects can lead to lower disc fractions in the dense cluster
centres (Guarcell et al., 2007; Pfalzner et al., 2014). Also, these effects
play a part, here we show that the main reason for the discrepancy in the
derived disk lifetimes is its sensitivity to cluster selection regarding age
and distance. We suggest that the stellar mass dependence might skew the
results towards short disk lifetimes. We find that restricting the sample to
nearby ($<$ 200 pc) clusters with an adequate balance between young and old
clusters leads to much longer median disk lifetimes of 5–10 Myr. The disk
fractions of the individual clusters in this sample are from two works (Michel
et al., 2021; Luhman, 2022). Within these two sources, the same method was
used to determine the disk fractions.

##  2 Correlation between cluster sample and disk lifetime

The work by Haisch et al. (2001) held the promise that additional data would
lead to a well-defined decline of the disk fraction with cluster age. However,
the more data became available, the wider became the spread. This spread is
often interpreted as being caused by external disk dispersal mechanisms. While
this explains the lower disc fraction in dense clusters (typical $n\approx$
104 pc-3 (Gutermuth et al., 2005)) ( see clusters indicated in green in Fig.
1, top left (Stolte et al., 2015; Vincke & Pfalzner, 2018; Concha-Ramírez et
al., 2021), it fails to account for the higher disc fractions in sparse
clusters (typical $n<$ 0.5 pc-3)). The black curve in Fig. 1 is based on
clusters with $n<$ 600 pc-3, where the effect of close stellar flybys on the
disc fractions is restricted to small areas close to the cluster centre. Many
of them have no O stars or much few than Upper Sco or UCL/LCC, excluding
external photo-evaporation as the main cause for the difference in disk
fraction. In total the environmental effect on the disc fraction should be
small ($\ll$5%) (Vincke & Pfalzner, 2018; Concha-Ramírez et al., 2021).
Furthermore, the disc fractions of older sparse associations and co-moving
groups are up to 25% higher (Fig. 1, top right, red data points) higher than
for the black curve. Thus density effects cannot be dominantly responsible for
this large difference.

Most clusters are detected as over-density relative to the background and
elevated infrared excesses. Young clusters are identifiable even if they
contain only a few hundreds or even tens of stars. When young clusters lose
most of their gas content at the end of the star formation phase (for example,
Lada & Lada, 2003; Kuhn et al., 2019), their size increases to 5 – 10 times
its initial value at ages 1 – 5 Myr. Thus, older clusters must contain $N>$
1000 stars to be detectable. This is why low-$N$ clusters older than 5 Myr are
missing from plots like Fig. 1 top right. Exceptions are the co-moving groups,
as they are discovered as moving in the same phase space independent of their
surface density.

Due to their initial compactness, clusters are identifiable at much larger
distances at young ages than later. Table 1 shows that studies including a
significant fraction of distant clusters arrive at shorter median disc
lifetimes. Even studies that include many old clusters, but all at large
distances, show this trend. Our central hypothesis is that limiting magnitude
at large distances introduces a bias towards brighter high-mass stars. As
high-mass stars tend to lose their discs earlier (see, for example, Carpenter
et al., 2006; Ribas et al., 2015), this skews the resulting disk lifetimes
towards shorter values.

Most young clusters contain 10 – 1000 times fewer stars than the older
clusters (Pfalzner et al., 2014). This makes the individual disc fraction of
young clusters less statistically significant in deriving the disk lifetime
than these of older clusters. In many studies, young clusters are over-
represented, while older clusters are under-represented. For example, in
Haisch et al. (2001) about 85% of clusters and all 69 clusters in Richert et
al. (2018) were $\leq$ 5 Myr. Table 1 shows examples of the fraction of
clusters $>$ 5 Myr in various studies. Studies with large fractions of young
clusters tend to derive short median disc lifetimes ($<$ 5 Myr) than those
with more long-lived clusters ($>$ 5 Myr).

Figure 1: Disk fraction vs cluster age. Top: Distinction between areas of
different density. Clusters generally used are shown in black, compact, dense
clusters in green, and sparse clusters in blue. The original fit by Haisch
(dashed black line). Middle: Fit using only clusters within 200 pc with the
age range 1 – 20 Myr equitably covered (red line and symbols). Bottom:
Clusters within 1000 pc. Symbols area proportional to the number of stars
considered and the colour representing the cluster distance.

##  3 Median disk lifetimes

To avoid these biases, we include nearby clusters ($<$ 200 pc) with an
equitable weight over the entire age range 1 – 20 Myr. Our sample contains
four clusters aged 1 - 3 Myr, five aged 3 – 8 Myr and four aged 8 – 20 Myr.
Fig. 1 top, right shows that the resulting decline of disk fraction with
cluster age (red solid line) is shallower than in studies mainly based on
young distant clusters (black line)(for values see Table 2). We fitted
different types of curves to the data. We find that the dependence of the disk
fraction on cluster age can be approximated by a one-parameter exponential
function of the form ${f_{D}=\exp(-t/7})$ with a median disc lifetime of 5.0
Myr or two-parameter exponential function of the form
$f_{D}=\exp(-0.25t^{0.8})$. These fits exclude the data of the even older
clusters $\sim$ 25 Myr old NGC 1960 and the 25 – 35 Myr old Alessi 30 (Galli
et al., 2021a) as it is unclear whether their disks are protoplanetary or
debris in nature. As an alternative, we also performed a Gaussian fit of the
corresponding disc life distribution (see accompanying research note). In this
case, we obtain a median disc lifetime of 6.5$\pm$1.5 Myr. Independent of the
applied method, the derived median disk lifetimes exceed considerably the
usually quoted 1 – 3 Myr. We emphasize that 25% of disks exist beyond 10 Myr
and $\approx$ 10% beyond 15 to 20 Myr.

The question is how statistically significant is this result. Everything else
being equal, the disk fractions of high-$N$ clusters are statistically more
significant. Fig. 1 bottom left shows the disc fraction as a function of
cluster age for the clusters within 1 kpc (see Table 2). However, here the
symbol area is approximately proportional to the number of stars, $N$,
considered in determining the disc fraction. It becomes apparent that the disc
fractions of Upper Sco and UCL/LCC are considerably more statistically
significant than those of the younger clusters. In total, there are 448 stars
surrounded by a protoplanetary disc in Upper Sco and UCL/LLC in (Luhman,
2022). Therefore it is improbable that some extreme outlier stars dominate by
their exceptionally long disc lifetimes. Thus the result of a median disc
lifetime exceeding 5 Myr seems to be highly statistically significant.

Besides these selection effects, some older studies assumed Upper Sco was
younger and had a lower disc fraction. Age estimates for Upper Sco range from
5 Myr to 12 Myr ( for example, Preibisch et al., 2002; Pecaut & Mamajek,
2016), however, with an increasing consensus towards an age of 10 – 12 Myr
(Feiden, 2016; Asensio-Torres et al., 2019). Before GAIA, significant
uncertainties in membership existed, especially in the cluster outskirts. Disk
fractions in the outer areas could be up to 3 times lower due to false
positives at that time Rizzuto et al. (2012). One strategy to avoid the
problem of false positives was to consider only the central cluster areas.
However, at least for high-mass clusters older than 3 – 5 Myr, this introduces
a bias toward lower disk fractions caused by external disk destruction
(Pfalzner et al., 2014). Nowadays, the false positive rate in the outskirts of
clusters is much lower. Recent disk fractions of Upper Sco are nearly twice as
high as the 11 % used in several older studies. Similarly, for the 15 – 20 Myr
old UCL/LCC region, disk fraction values increased from 1% – 3% to 9 %
nowadays.

Why does restricting to nearby clusters with an even spread in cluster ages
lead to a longer disc lifetime? Michel et al. (2021) found the disk fraction
of Upper Sco to be very similar to that of co-moving groups of similar age.
They argue that the low-UV radiation in both samples is the reason for the
similarity in disk fractions. In the following, we reason that distance rather
than similar radiation levels might lead to disc fractions being similar for
these otherwise quite different environments.

##  4 Median disk lifetime: a question of stellar mass

Many studies found that disk fractions to be lower for high- than low-mass
stars (e.g., Carpenter et al., 2006; Roccatagliata et al., 2011; Yasui et al.,
2014; Ribas et al., 2015; Richert et al., 2018). Here we hypothesise that this
stellar mass dependence of disc fractions is partly responsible for the wide
spread in disc fractions at any given age. Most stars are of low mass (M- and
K-type); however, observations of more distant clusters or at low sensitivity
suffer from limiting magnitude. Thus mean stellar mass is higher as low-mass
stars are under-represented in these samples. This bias towards higher stellar
mass affects the derived disc lifetimes directly. As high-mass stars lose
their disks faster(for example, Ribas et al., 2015), distant cluster disk
fractions are systematically lower. Consequently, the derived disc lifetimes
are shorter.

We test this hypothesis by plotting the disk fractions as a function of
stellar age colour-coded by cluster distance (Fig. 1 bottom left). A strong
correlation between the slope of the decline in disk fraction with cluster
distance becomes apparent. Obviously, current median disk lifetimes suffer
from a strong selection effect connected to cluster distance. Here cluster
distance is only a crude proxy for missing out on low-mass stars in the more
distant samples but is sufficient to demonstrate the problem.

The mean density of distant clusters like Arches or Trumpler 14 clusters is
about a thousand times higher than in clusters like the ONC. These dense
clusters are located at distances $>$ 2000 pc. Therefore they are strongly
affected by limiting magnitude; The derived very low disc fractions usually
apply only to stars with masses $>$ 1 $M_{\odot}$. However, environmental
effects should also apply to these clusters (Mann et al., 2014; Vincke &
Pfalzner, 2016; Ansdell et al., 2017). However, determining the relative
importance of mass dependence vs environment remains a challenging task.

Upper Sco has a disk fraction of 5%${}^{+4\%}_{-3\%}$ for B7–K5.5-type stars
(Luhman, 2022) compared to 22%$\pm$0.02 for low-mass stars (M3.7-M6). Similar,
for the UCL/LLC, the disk fraction is 0.7%${}^{+0.06}_{-0.04}$ for higher mass
stars compared to 9%$\pm$1% for low-mass stars. The statistical significance
of these data is quite high, even when separated into mass bins. For M3.75-M6
stars (22%) it is based on 633 objects, for K6-M3.5 stars (18%) on 311
objects, and for B7–K5.5-type stars (5%${}^{+4\%}_{-3\%}$) on 76 objects in
(Luhman, 2022). So even for the high-mass star, the statistical sample is as
large as many young low-mass clusters’ total population. The statistical
significance is even higher for the UCL/LLC data, with sample sizes of 2488,
725 and 452 in the equivalent stellar mass bins.

Figure 2: Disk fraction vs. cluster age. Top: colors indicating different
cluster distances. Middle: colors indicating different limiting magnitudes.
Bottom: Fits for mass-dependent disc lifetimes based on the high-significance
values of Upper Sco and UCL/LCC assuming an initial disc fraction of 100%
(solid line) and 75% dashed line

.

This dependence of the disk fractions on stellar mass translates into shorter
disc lifetimes for high- than low-mass stars. Figure 2 shows a simple
extrapolation based on the Upper Sco and UCL/LLC data, assuming an initial
disk fraction of 100%. Basing the curves on just three points is not ideal
(the third one being the assumed primordial disc fraction, but as shown in
Fig. 1, bottom, the fit for the total disk fraction holds for additional
clusters. Nevertheless, it illustrates the critical point of the mass-
dependence of the decline in disc fraction. Using Fig. 2 as a first
indication, we expect that 50% of the low-mass stars still have a disk at
$\approx$ 5 Myr. By contrast, only $<$ 20% of the high-mass stars retain their
disks at that age. Half of the high-mass stars have already lost their disks
at $\approx$ 2 Myr. Thus the median disk lifetime of high- and low-mass stars
seems to differ at least a factor of two.

There is considerable uncertainty of the zero-age disk frequency (Michel et
al., 2021). If all stars are initially surrounded by disks, the median disk
frequency for the low-mass stars would be 5 – 7 Myr, whereas for initial disk
frequencies of 80%, the median disk lifetime increases to 6 – 10 Myr. (see
Fig. 2 bottom). Here again the higher values are obtained when performing a
Gaussian fit the corresponding disc life distribution.

The mass dependence disk lifetime could be determined from the disk fractions
for the different stellar masses. Unfortunately, there is a lack of
statistically meaningful data for high stellar masses. Fig. 3 illustrates the
general trend based on the Upper Sco and UCL/LCC data. Both scenarios (initial
disk frequency of 100% and 80%) show the same overall trend: For M – K stars,
the median disk lifetime is basically the same, but for higher mass stars, it
is considerably lower. However, where precisely the decline in disk lifetimes
happens remains uncertain. This relation requires urgently further
observational investigation.

Why do high-mass stars lose their disks so much earlier than low-mass stars?
One answer might lie in their higher efficiency in accreting material and
photo-evaporation. Wilhelm & Portegies Zwart (2021) find that stars with
masses exceeding 0.8 $M_{\odot}$ have shorter lifetimes due to these two
effects; nevertheless, lifetimes up to 15 Myr are still possible for all host
star masses up to 2 $M_{\odot}$. The critical role of accretion is supported
by observations finding that the lowest M stars still that retain a disk at
ages $\approx$ 8 – 10 Myr also show moderate accretion levels Venuti et al.
(2019).

##  5 From the diversity of disk life times to that of planets

The biggest surprise in exoplanet observations is the immense diversity in
planets and planetary systems (Howard, 2013; Gaudi et al., 2021). Suggested
causes are, among others, differences in the disk mass and disk mass profile
(Kokubo & Ida, 2002), the metallicity of the stars (Petigura et al., 2018),
the location of rings in discs (van der Marel & Mulders, 2021), and the type
of cluster environment (Bate, 2018; Vincke & Pfalzner, 2018; Winter et al.,
2020).

The variation in individual disk lifetime has also been suggested to influence
the type of planet that forms and the architecture of the planetary systems
(Carpenter et al., 2005; Luhman & Mamajek, 2012; Ribas et al., 2015). If that
holds, the stellar mass dependence of the disc lifetime should result in
differences in the planets as a function of stellar mass.

Figure 3: Top: Median disk lifetime as function of stellar mass assuming 100 %
initial disc fraction (dashed lines,) 80% initially disc fraction (solid
lines) and assuming a lower Upper Sco age of 8 Myr instead of 11 Myr (dotted
lines).The orange lines show the general trend including the uncertainty for
higher mass stars. Bottom: Correlation between disk lifetime and properties of
planetary system.

Disk masses scale quasi linearly with stellar mass (e.g., Andrews, 2020). Thus
high-mass stars have a much larger gas and dust reservoir for planet
formation. Therefore it is not surprising that main-sequence FGK stars host
more larger planets than low-mass stars (Howard et al., 2012; Sabotta et al.,
2021). However, the situation is different for smaller planets. M dwarfs host
about a factor of three more small planets (1.0–2.8 R⊕ ) than main-sequence
FGK stars (for example, Mulders et al., 2021). However, the mass is not simply
redistributed into more, smaller planets. Surprisingly, the average heavy-
element mass decreases with increasing stellar mass from 7 M⊕ for M stars to 5
M⊕ for G and K stars and 4 M⊕ for F stars. Thus despite M star disks
containing ten times less mass, they are nearly 20 times as efficient than F
stars in converting the disk’s heavy-element content into planetary material.
The higher heavy-element content also corresponds to a higher fraction of
stars with planetary systems for low-mass stars (Yang et al., 2020; He et al.,
2021).

The strong dependence of the disk lifetime on stellar mass may explain the
high planet-formation efficiency in low-mass stars. High-mass stars seem to
produce their high-mass gas giants on timescales shorter than 3 Myr while
failing to form additional low-mass planets. By contrast, low-mass stars form
large planets to a lower degree; however, their long disk lifetimes allow for
the formation of many small planets. In a way, slow but steady beats fast and
short.

Forming low-mass planets on time scales $>$ 5 Myr lowers the hurdles for the
standard accretion model. However, explaining why high-mass stars form more
massive planets on considerably shorter time scales remains a significant
challenge. A possible explanation is that their higher disk masses make their
disks more prone to gravitational instabilities. However, as $M_{d}/M_{s}\sim$
const independent of stellar mass, the parameter $Q$ that describes the
stability of disks should be the same for high and low mass stars. Thus
unstable disks are at least not a straightforward explanation. Thus planet
formation around high-mass stars remains an open question.

##  6 Summary and Conclusion

We investigated the role of cluster sample selection on derived median disc
lifetimes. We find that samples with a large fraction of distant, young
clusters ($>$ 200 pc; $<$ 5 Myr) tend to derive short disc lifetimes (1 – 3
Myr). Samples including higher fractions of nearby, older clusters arrive at
higher disc lifetimes ($>$ 5 Myr). Restricting to clusters closer than 200 pc
aged between 1 and 20 Myr, we obtain a median disk lifetime of 5 to 10 Myr.

One main reason for this discrepancy is that distant clusters are affected by
limiting magnitude. Therefore, the disc fraction of, on average, higher mass
stars is determined in distant clusters. As the disk lifetime of high-mass
stars is shorter than for low-mass stars, the disc fractions of more distant
clusters seem lower due to this selection effect. We conclude that disc
lifetimes derived from samples including many distant clusters ($>$ 200 pc)
represent mostly high-mass stars. Indeed, if we only consider the high-mass
stars in the sample limited to distances $<$ 200 pc, we recover a disk
lifetime of only 2 – 4 Myr.

However, these disc lifetimes are not representative of most of the stars.
Low-mass stars have a median disc lifetime of 5 – 10 Myr. The actual value
depends mainly on the assumed initial disc fraction. If all stars are
surrounded by disks at cluster ages $t_{c}$ = 0 Myr, the median disc lifetime
is 5– 6 Myr. However, some stars seem to be born diskless or lose their disk
extremely rapidly to planet formation. An initial disk fraction of 70%– 80 %
would increase the disk dissipation times for low-mass stars to 8 – 10 Myr and
that of high-mass stars to 4 – 5 Myr.

For low-mass stars, the median disk lifetime of 5 – 10 Myr significantly
relaxes the temporal constraints on planet formation. These long disk
lifetimes allow for sufficient time for planets to form via accretion. The
diversity of disk lifetimes might influence the structure of the emerging
planetary system. It could be responsible for low-mass stars having
considerably higher efficiency in using the heavy-element content in their
disk for planet formation. Despite having considerably lower disk masses to
start with, these low-mass stars produce a larger number of lower-mass
planets. The real challenge remains to explain how high-mass stars can form
planets on such a short timescale.

Currently, the effect of the environment on the disk lifetime is still not
quantifiable. Only disk fractions for high-mass star clusters have been
determined for dense clusters due to their general large distances ($>$ 2000
pc). A comparison is difficult even for those high-mass stars as the initial
disc fraction in these dense clusters is unknown. Its uncertainty is very
high, as none of the measured disc fractions exceed 32%.

Generally, determining the initial disc fraction is the next step to determine
disc lifetimes accurately. This includes potential dependencies on stellar
mass, cluster density and binarity.

We thank the referee for the constructive report and useful suggestions that
have significantly improved our manuscript. We would like to thank K. Luhman
for helpful advice on interpreting his results on the disk fractions in Upper
Sco and UCL/LCC.

Table 2: Disk fractions Identification | $d$ | Age | $N_{stars}$ | $f_{d}$ | Limit | Median mass | log($\rho_{c})$ | Source
---|---|---|---|---|---|---|---|---
| pc | Myr |  |  |  | [$M_{\odot}$] | [$M_{\odot}$/pc3] |
$d\ll$ 200 pc |  |  |  |  |  |  |  |
Alessi 30 | 108 | 30 | 162 | 0.049111possibly debris disk fraction | 0.04 $M_{\odot}$ |  |  | a)
UCL/LLC | 150 | 15–20 | 3665 | 0.09 |  | 0.15 | -0.85 -(-1.05) | b)
32 Ori | 95 | 15–20 | 160 | 0.07 |  | 0.15 |  | f1
Upp Sco | 145 | 10–12 | 1774 | 0.22/0.20 | 0.01 $M_{\odot}$ | 0.15 | -0.59 | b, c)
Lupus -off cloud | 160 | 10–12 | 60 | 0.21 $\pm$ 0.06 | 0.05 $M_{\odot}$ |  |  | l)
$\eta$ Cha | 94 | 8–14 | 40 | 0.28/33 |  |  |  | d)
TW Hya | 56 | 7–13 | 56 | 0.25/0.30 |  |  |  | d)
Lupus-on cloud | 160 | 6 | 30 | 0.63 $\pm$ 0.04 | 0.05 $M_{\odot}$ |  |  | l)
CrA | 152 | 5 | 146 | 0.23 $\pm$ 0.4 | 0.04 $M_{\odot}$ |  |  | d)
$\epsilon$ Cha | 101 | 5 (3–8) | 90 | 0.5/0.3222The disk fraction is much higher in the center than the outskirts ($>$10 pc) of $\eta$ Cha |  |  |  | d, e)
Lupus | 158 | 2.6–3.1 | 158 | 0.50/0.53 | 0.03 $M_{\odot}$ |  |  | a, d)
Cham I | 188 | 1.7 | 183 | 0.44 | 6 $<$G $<$20 |  |  | f)
Cham II | 197 | 1.7 | 41 | 0.76 | G12-G18 |  |  | f)
Taurus | 128-196 | 1–2 | 137 | 0.49 | 0.05 $M_{\odot}$ |  |  | i)
Ophiuchus | 139 | 1–2 | 420 | 0.62 |  |  |  | i)
200 pc $<d<$ 500 pc |  |  |  |  |  |  |  |
25 Orionis | 330 | 8.5 $\pm 1.5$ | 26 | 0.09 |  |  |  | g)
Ori 1a | 355 | 8 | 811 | 0.08 | 0.1 $M_{\odot}$ |  | -0.54 | a1)
$\gamma$ Vel | 345 | 7.5 | 125 | 0.06 | 0.2 $M_{\odot}$ |  |  | i)
$\lambda$ Ori | 414 | 5 | 43 | 0.19 |  |  |  | i)
OriOB1b | 414 | 5 | 278 | 0.15 | 0.04 $M_{\odot}$ |  | 0.55 | i)
IC 348 | 310 | 2-3 | 310 | 0.50/0.40 |  |  | 2.2 | d)
$\sigma$ Ori | 414 | 2.5 | 71 | 0.36 |  | 0.3 $M_{\odot}$ |  | i)
NGC2068/NGC2071 | 400 | 1-3 | 67 | 0.54 |  |  |  | j)
Berkley59 | 400 | 1.8 | 201 | 0.50 | 0.1 $M_{\odot}$ | 0.78 |  | k)
Serpens South | 415 | 1.8 | 26 | 0.58 | 0.1 $M_{\odot}$ | 0.3 |  | k)
ONC Flank | 414 | 1.7 | 236 | 0.43 | 0.13 $M_{\odot}$ | 0.52 | 1.3 | k)
OMC | 414 | 1.5 | 181 | 0.45 | 0.09 $M_{\odot}$ | 0.32 |  | k)
L1630N | 400 | 1.5 |  | 0.97 |  |  |  | l)
Lynds1641 | 400 | 1.5 |  | 0.51 |  |  |  | m)
NGC1333 | 320 | 1.0 | 73 | 0.81 |  |  |  | j)
Flame/NGC2023 | 414 | 0.8 | 142 | 0.71 | 0.1 | 0.38 $M_{\odot}$ |  | k)
Serpens | 425 | 0.5 | 137 | 0.75 |  |  |  | j)
NGC2024 | 415 | 0.3 |  | 0.85 |  |  | 2.424 | a1)
500 pc $<d<$ 1000 pc |  |  |  |  |  |  |  |
NGC 7160 | 900 | 11 $\pm$ 1 | – | 0.04 $\pm$ 0.03 |  |  |  | b1)
CepOB3b-East | 700 | 3.5 |  | 0.32 |  |  |  | x)
CepOB3b-West | 700 | 3.5 |  | 0.50 |  |  |  | y)
NGC2264 | 760 | 3.2 | 324 | 0.52 |  |  |  | c1)
| 751 | 3.1 |  | 0.38 |  |  |  | j)
Trumpler37 | 900 | 2.6 |  | 0.49 |  |  |  | j)
CepC | 700 | 2.2 | 59 | 0.44 | 0.1 $M_{\odot}$ | 0.47 |  | k)
CepA/A | 700 | 2.0 | 77 | 0.38 | 0.1 $M_{\odot}$ | 0.43 |  | k)
MonR2 | 830 | 1.7 | 208 | 0.64 | 0.09 $M_{\odot}$ | 0.47 | 1.72 | k)
LkH$\alpha$101 | 510 | 1.5 | 140 | 0.56 | 0.1 $M_{\odot}$ | 0.56 |  | k)
L988e | 700 | 1.5 |  | 0.79 |  |  |  | d1)
Continued on next page |  |  |  |  |  |
Table 2: continued Identification | $d$ | Age | $N_{stars}$ | $f_{d}$ | Limit | Median mass | log($\rho_{c})$ | Source
---|---|---|---|---|---|---|---|---
| pc | Myr |  |  |  | [$M_{\odot}$] | [$M_{\odot}$/pc3] |
CepA/C | 700 | 1.4 | 86 | 0.65 | 0.1 $M_{\odot}$ | 0.38 |  | k)
RCW36 | 700 | 0.9 |  | 0.81 | 0.1 $M_{\odot}$ | 0.35 |  | k)
W40 | 500 | 0.8 |  | 0.79 | 0.1 $M_{\odot}$ | 0.53 |  | k
1000 pc $<d<$ 2000 pc |  |  |  |  |  |  |  |
NGC6231 | 1585 | 4 |  | 0.05 |  |  |  | n)
NGC2282 | 1650 | 3.5 |  | 0.58 |  |  |  | o)
NGC7129 | 1260 | 3 |  | 0.33 $\pm$ 0.22 |  |  |  | q)
W3Main | 1950 | 3 |  | 0.07 |  |  |  | r)
NGC2244 | 1880 | 3 | 570 | 0.445 |  |  | 1.03 | e1)
NGC2362 | 1480 | 2.5 |  | 0.12 |  |  |  | k)
M8 | 1300 | 2.3 |  | 0.41 |  |  |  | k)
AFGL333 | 2000 | 2.0 |  | 0.55 $\pm$ 0.5 |  |  |  | s)
NGC 6611 | 1995 | 2.0 |  | 0.59 |  |  |  |
| 1750 | 1.2 |  | 0.34 |  |  | 1.45 | u)
Pismis 24 | 1700 | 1.85 |  | 0.33 |  |  |  | t)
Cyngus OB2 | 1450 | 1.5 |  | 0.29 |  |  | 1.61 | u)
M 17 | 2000 | 1.1 | 35 | 0.6 | 0.12 $M_{\odot}$ | 3.68 |  | k)
Sh2-106 | 1400 | 0.8 | 92 | 0.53 $\pm$ 0.1 | 0.13 $M_{\odot}$ | 0.6 |  | k)
NGC 6530 | 1300 | 0.7 |  | 0.20 |  |  |  | v)
$d>$ 2000 pc |  |  |  |  |  |  |  |
Trumpler 15 | 2360 | 8.0 |  | 0.021 | 1 $M_{\odot}$ |  |  | w)
Bochum 1 | 2800 | 5.0 |  | 0.086 | A -B |  |  | w)
Quintuplett | 8000 | 4.0 | 766 | 0.04 | A - B |  | 3.7 | z)
Trumpler 16 | 2700 | 3.0 |  | 0.069 | 1 $M_{\odot}$ |  |  | w)
Trumpler 14 | 2800 | 2.0 |  | 0.097 | 1 $M_{\odot}$ |  | 4.3 | w)
Arches | 8000 | 2.5 |  | 0.092 | A - B |  | 5.6 | z)
NGC 3603 | 3600 | 1.0 |  | 0.27 |  |  | 5.0 | z)
TTC | 2700 | 0.1 |  | 0.32 | 1 $M_{\odot}$ |  |  | w)

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# Optimizing optical tweezing with directional scattering in composite
microspheres

R. Ali<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio de
Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brasil  F. A.
Pinheiro  Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa
Postal 68528, Rio de Janeiro, RJ, 21941-972, Brasil  F. S. S. Rosa  Instituto
de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de
Janeiro, RJ, 21941-972, Brasil  R. S. Dutra  LISComp-IFRJ, Instituto Federal
de Educação, Ciência e Tecnologia, Rua Sebastião de Lacerda, Paracambi, RJ,
26600-000, Brasil  P. A. Maia Neto  Instituto de Física, Universidade Federal
do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ, 21941-972, Brasil

###### Abstract

Trapping of microspheres with a single focused laser beam is usually limited
to materials with relative refractive indexes slightly larger than one. We
show that directional light scattering can be employed to optically trap high-
index materials. For this purpose, we propose a material platform to achieve
zero backward scattering (ZBS), also known as the first Kerker condition, in a
composite media containing spherical inclusions of silica embedded in a SiC
microsphere. By tuning the volume filling fraction of inclusions and the
microsphere radius, stable trapping can be achieved, provided that ZBS is
combined with the condition for destructive interference between the fields
reflected at the external and internal interfaces of the microsphere when
located at the focal point. We show that our proposal also holds even in the
presence of a significant amount of spherical aberration, which is a common
condition in most optical tweezers setups. In this case, achieving ZBS is
essential for trapping high-index materials.

##  I Introduction

Optical tweezers - laser traps for neutral microscopic particles - are a
powerful optical tool with many applications in physics and biology ashkin1986
; ashkin2006 . In biology, optical tweezers allowed for pioneering
quantitative measurements of basic interactions in living cells fazal2011 . In
physics, optical tweezers have been employed to tackle important fundamental
problems, such as the experimental implementation of the Szilard’s demon
toyabe2010 , the first experimental proof of Landauer’s principle brut2012
and femtonewton force measurements that open the way for probing non-trivial
geometry effects in the double-layer and Casimir interactions diney .

In a typical optical tweezer setup, a dielectric microsphere inside a water-
filled sample chamber is illuminated by a single laser beam brought to a
diffraction limited focal spot. Whereas reflection gives rise to radiation
pressure that pushes the microsphere along the overall propagation direction,
refraction provides a restoring force pointing to the focal point if the
sphere relative refractive index $m$ is larger than one. Trapping is achieved
provided $m$ is only slightly larger than one, so that refraction dominates
over reflection. This condition imposes a severe limitation in many
applications of practical interest. For instance, in Casimir force
measurements with optical tweezers, the magnitude of the interaction usually
increases with the sphere refractive index, and a larger experimental signal
would be obtained if optical trapping of spheres with large refractive indexes
was possible.

In order to circumvent this limitation we explore novel mechanisms to control
the radiation scattering pattern, which is now possible due to progress in the
field of metamaterials. Specifically, we introduce the concept of Zero
Backward Scattering (ZBS), also known as first Kerker condition, in the field
of optical trapping. The first and the second Kerker conditions, the latter
being related to Zero Forward Scattering (ZFS), were put forward in 1983, when
it was theoretically shown that in a magnetic sphere coherent effects between
electric and magnetic dipoles may lead to strongly asymmetric radiation
patterns ker . For many years the observation of these effects for optical
frequencies was considered to be impossible due to the fact that the relative
magnetic permeability of natural media is unity in this frequency range.
However, with the advent of metamaterials, which allowed for optical magnetism
monticone , this limitation has been overcome and achieving directional
scattering and Kerker conditions have attracted considerable attention in
recent years. Indeed, several experimental realizations to obtain directional
scattering exist, such as single GaAs Si , silicon fu2013 ; ge , or dielectric
nanoparticles geffrin2012 , as well as gold nanoantennas alaee2015 . In
addition, different approaches toward directional and/or anomalous scattering
patterns have been proposed, such as exploiting interferences among dipolar
and quadrupolar resonances li2015 , using gain xie2015 , core-shell liu2012
and highly refractive nanoparticles kuznetsov2016 , or designing pyramidal
nanostructures to excite magnetic resonances rodriguez2014 .

In the present paper we propose a new material platform to achieve ZBS based
on non-magnetic composite media and apply it to facilitate and optimize
optical tweezing. Using extended Maxwell-Garnett effective medium and Mie
theories, we demonstrate that one can achieve nearly ZBS for some values of
the the filling fraction of inclusions. For concreteness, we consider a
microsphere of ${\rm SiC}$ with ${\rm SiO_{2}}$ inclusions at the typical
operation wavelength of optical tweezers, $1064\,{\rm nm}.$ At such
wavelength, an homogeneous microsphere of ${\rm SiC}$ cannot be optically
trapped in water due to its large refractive index $n_{\rm SiC}=2.57$. We show
that one can circumvent this limitation by adding inclusions that allow for a
drastic reduction of backscattering radiation, hence optimizing optical
trapping. We demonstrate that by fulfilling the first Kerker condition one can
not only trap high refractive index spheres that cannot be trapped in general,
but also enhance trapping stability. Besides, we prove that this result is
even more important if one considers the spherical aberration caused by
refraction at the interface between the glass slide and the water-filled
sample chamber, which typically occurs in the vast majority of cases of
practical interest. Altogether our results unveil the role of backscattering
in optical tweezing and pave the way for the design of new experimental
trapping devices.

This paper is organized as follows. Section II is devoted to the description
of the methodology, where Mie and extended Maxwell-Garnett theories are
briefly presented. In Sec. III the conditions for optical tweezing are
discussed and our main results are presented and analyzed. Finally, in Sec. IV
we summarize our findings and conclusions.

##  II Methodology

###  II.1 Mie Theory and Kerker conditions

When an electromagnetic plane wave (vacuum wavelength $\lambda_{0}$)
illuminates a spherical particle (radius $r$ and relative refractive index
$m$), scattering and absorption processes can be described in terms of the
spherical multipole decomposition using Mie theory bohren . The scattered
field can be written as a function of the Mie coefficients $a_{\ell}$ and
$b_{\ell}$, which correspond to the electric and magnetic multipoles,
respectively. Here the index ${\ell}$ is used to denote the ${\ell}^{th}-$
order spherical harmonic channel. The Mie coefficients are functions of the
size parameter $x=kr,$ where $k$ is the wavenumber in the host medium bohren :

$a_{\ell}=\dfrac{m\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-\mu\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{m\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-\mu\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}$
(1)

$b_{\ell}=\dfrac{\mu\psi_{\ell}(mx)\psi^{\prime}_{\ell}(x)-m\psi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}{\mu\psi_{\ell}(mx)\xi^{\prime}_{\ell}(x)-m\xi_{\ell}(x)\psi^{\prime}_{\ell}(mx)}$
(2)

where $\psi_{\ell}$, $\xi_{\ell}$ are Riccati-Bessel functions DLMF25.12 , and
$\mu$ is the magnetic permeability of the sphere. The extinction, absorption
and scattering cross-section efficiencies read bohren

$Q_{ext}=Q_{abs}+Q_{scat},$
$Q_{abs}=\frac{2}{x^{2}}\sum_{\ell=1}^{\infty}{(2{\ell}+1)(Re{[a_{\ell}]}-|a_{\ell}|^{2}+Re[b_{\ell}]-|b_{\ell}|^{2}}),$
$Q_{scat}=\frac{2}{x^{2}}\sum_{\ell=1}^{\infty}{(2{\ell}+1)(|a_{\ell}|^{2}+|b_{\ell}|^{2}}).$

Information on the directionality of the scattered radiation can be obtained
in terms of $a_{\ell}$ and $b_{\ell}$ by defining the differential scattering
cross-sections in the forward ($\theta=0$) and backward ($\theta=\pi)$
directions bohren :

$\frac{dQ}{d\Omega}\bigg{|}_{\theta=0}=Q_{f}=\frac{1}{x^{2}}\left|\sum_{\ell=1}^{\infty}{(2\ell+1)(a_{\ell}+b_{\ell})}\right|^{2}$
(3)
$\frac{dQ}{d\Omega}\bigg{|}_{\theta=\pi}=Q_{b}=\frac{1}{x^{2}}\left|\sum_{\ell=1}^{\infty}{(2{\ell}+1)(-1)^{\ell}(a_{\ell}-b_{\ell})}\right|^{2}.$
(4)

The conditions for ZBS and ZFS were established by Kerker et al. ker  for
magnetodielectric spheres as a result of interferences between magnetic and
electric scattering resonances ker . When the electric and magnetic multipoles
with the same amplitudes oscillate in-phase so that $a_{\ell}=b_{\ell}$, ZBS
is possible as it can be seen in Eq. 4 (first Kerker condition); this occurs
provided $\epsilon$ = $\mu$. On the other hand, when the electric and magnetic
multipoles oscillate out-of-phase, that is $a_{\ell}=-b_{\ell}$, ZFS may arise
as Eq. 3 reveals (second Kerker condition); this occurs when the condition
$\epsilon=\frac{4-\mu}{2\mu+1}$ is satisfied in the quasistatic limit. Note
that a complete vanishing of forward scattering cannot occur for any
noninvisible passive system since, according to the optical theorem, this
would strictly imply in vanishing of absorption and scattering cross sections
alu2010 ; lee2017 . However, there is no such restriction for ZBS, which is
our interest here, while nearly zero forward scattering is possible for core-
shell nanoparticles Lee . In the next subsection we shall demonstrate that ZBS
can be achieved in composite, non-magnetic dielectric microspheres.

Figure 1:  Real (red) and imaginary (blue) parts of the composite effective
refractive index $n_{\rm emg}=\sqrt{\epsilon_{\rm emg}\mu_{\rm emg}},$
calculated with the extended Maxwell Garnett theory, versus volume filling
fraction. The inset illustrates the composite made of ${\rm SiO_{2}}$
spherical inclusions (radius $8\,{\rm nm}$) distributed inside a ${\rm SiC}$
sphere.

###  II.2 Extended Maxwell-Garnett Theory

When electromagnetic radiation propagates through a host medium with
inclusions (for instance, a porous medium) much smaller than the
characteristic wavelength, we can describe such a material by an effective
medium theory Choy_EMT . Two of the most popular effective medium theories are
the so called Maxwell-Garnett (MG) theory max ; max1  and its generalization,
the Extended Maxwell-Garnett (EMG) theory Doyle_89 ; rup . In the former one
considers small spherical particles of radius $a$ and permittivity
$\epsilon_{\rm i}$ embedded in host medium of permittivity $\epsilon_{\rm h}$,
with a volume filling fraction $f$. Then, MG theory consists essentially in
invoking the Lorentz-Lorenz formula bohren  with the static polarizability of
a dielectric sphere, resulting in rup

$\epsilon_{\rm mg}=\epsilon_{\rm h}\dfrac{\epsilon_{\rm
i}(1+2f)+2\epsilon_{\rm h}(1-f)}{\epsilon_{\rm i}(1-2f)+\epsilon_{\rm
h}(2+f)}.$ (5)

The MG effective permittivity (5) is independent of the size of the
inclusions, which however must satisfy the condition
${\rm{\it{a}}}\ll\lambda_{0}.$

Figure 2: (a) Backscattering efficiency $Q_{b}$ versus volume filling fraction
$f,$ showing almost zero backscattering efficiency around $f=0.5$ for a
composite microsphere of radius $r=1825\,{\rm nm}.$ (b) Real and imaginary
parts of the dipolar Mie coefficients versus filling fraction. The vertical
line at $f=0.5$ shows the intersection of ${\rm Re}(a_{1})$ (solid red) and
${\rm Re}(b_{1})$ (solid blue), which happens to be quite close to the
intersection of ${\rm Im}(a_{1})$ (dashed red) and ${\rm Im}(b_{1})$ (dashed
red).

The extended Maxwell Garnett (EMG) theory is still based on the Lorentz-Lorenz
formula, but now the (electric) dipolar polarizability is given by the full
electrodynamical expression, leading to rup

$\epsilon_{\rm emg}=\epsilon_{\rm
h}\dfrac{x^{3}+3ifa_{1}}{x^{3}-\frac{3}{2}ifa_{1}}$ (6)

where $a_{1}$ is the electric dipolar Mie coefficient (1), and
$x=\sqrt{\epsilon}_{h}\omega a/c$ is the size parameter within the host
medium. The EMG is most useful when we have $x\ll 1$ (so the inclusions may be
considered as dipoles), but the size parameter within the inclusions
$y=\sqrt{\epsilon}_{i}\omega a/c$ is not necessarily small (so the scattering
has to be solved with the Mie theory). By an analogous reasoning, the EMG also
gives rise to an effective magnetic permeability rup

$\mu_{\rm emg}=\mu_{h}\dfrac{x^{3}+3ifb_{1}}{x^{3}-\frac{3}{2}ifb_{1}}$ (7)

where $b_{1}$ is the magnetic dipolar Mie coefficient (2). In the framework of
EMG theory, even strictly non-magnetic materials may give rise to an effective
permeability, as long as $b_{1}$ is non negligible. We should also stress that
these results for $\epsilon_{\rm emg}$ and $\mu_{\rm emg}$ are not necessarily
restricted to low filling fractions, provided the positions of the inclusions
are uncorrelated multipolescattering . We would like to apply these ideas to
the scattering of light from a microsphere of SiC (the host), with nanospheres
of ${\rm SiO_{2}}$ (the inclusions) embedded in it, as shown in the inset of
Fig. 1. The effective refractive index $n_{\rm emg}=\sqrt{\epsilon_{\rm
emg}\mu_{\rm emg}}$ as a function of volume filling fraction $f$ is also shown
in Fig. 1. We take $\lambda_{0}=1064\,{\rm nm}$, the typical wavelength in
most optical tweezers setups. The radius of the microsphere is $r=1825\,{\rm
nm}$ while the nanospheres radii are $a=8\,{\rm nm}$, and the composite is
immersed in water. In Fig. 2a the backscattering cross section $Q_{b}$ is
plotted versus volume filling fraction, and we see a big dip right at $f=0.5$.
This is partially explained by the plots in Fig. 2b, where it is shown that
around $f=0.5$ the $a_{1}$ and $b_{1}$ Mie coefficients of the composite
microsphere nearly coincide, thus fulfilling the first Kerker condition for
${\ell}=1$ and therefore reducing the backscattering. It is possible to show
that subsequent electric and magnetic Mie coefficients also meet the first
Kerker condition around $f=0.5$, leading to the stark suppression of
backscattering shown in Fig. 2(a).

###  II.3 Mie-Debye theory of optical tweezers

Optical trapping of microspheres with a single laser beam is possible only by
employing high numerical aperture objectives well beyond the paraxial regime.
Thus, an accurate theoretical description must be based on a nonparaxial
modelling of the trapping beam. The Mie-Debye theory of optical tweezers epl
combines the electromagnetic generalization RichardsWolf  of Debye’s scalar
model for aplanatic focused beams with Mie scattering theory. Each plane wave
component of the incident beam leads to a scattered field component which is
easily related to standard Mie-scattering formulae with the help of Wigner
finite rotation matrix elements $d^{\ell}_{m,m^{\prime}}(\theta)$ Edmonds ,
where $\theta$ is the angle between each wavevector and the $z$-axis.

Since the optical force always points towards the beam axis $z$ at any
position on the $xy$ plane, it is sufficient to analyze the optical force
component $F_{z}$ along the $z$-axis in order to discuss the requirements for
three-dimensional trapping. When the sphere center and the trapping beam focal
point are aligned along the $z$-axis, the multipole coefficients for the
incident laser beam are given by

$G_{\ell}=\int_{0}^{\theta_{0}}d\theta\sin\theta\sqrt{\cos\theta}e^{-\gamma^{2}\sin^{2}\theta}\,\exp(ikz\cos\theta)\,d_{1,1}^{\ell}(\theta)$
(8)

where $z$ is the position of the sphere center with respect to the focal
point, and $\gamma$ is the ratio of the objective focal length to the laser
beam waist at the objective entrance port. The angular aperture
$\theta_{0}=\sin^{-1}(\mbox{NA}/n)$ is defined by the objective numerical
aperture (NA).

It is convenient to define the dimensionless force efficiency ashkinbio

$Q_{z}=\frac{F_{z}}{nP/c}$

where $P$ is the laser beam power at the sample region. We write the force
efficiency as the sum of two contributions:

$Q_{z}=Q_{e}{}_{z}+Q_{s}{}_{z}.$ (9)

The extinction term epl

$Q_{e}{}_{z}=-\frac{4\gamma^{2}}{A}\mbox{Im}\sum^{\infty}_{\ell=1}(2\ell+1)(a_{\ell}+b_{\ell})G_{\ell}\left(\dfrac{\partial
G_{\ell}}{\partial(kz)}\right)^{*}$ (10)

accounts for the rate at which linear momentum is removed from the incident
laser beam. The factor $A=1-\exp(-2\gamma^{2}\sin^{2}\theta_{0})$ is the
fraction of the trapping beam power that fills the objective entrance
aperture.

The term $Q_{s}{}_{z}$ in (9) represents the negative of the linear momentum
rate carried away by scattered field:

$\displaystyle
Q_{s}{}_{z}=-\frac{8\gamma^{2}}{A}\mbox{Re}\sum^{\infty}_{\ell=1}\left[\frac{\ell(\ell+2)}{(\ell+1)}(a_{\ell}a_{\ell+1}^{*}+b_{\ell}b_{\ell+1}^{*})G_{\ell}G_{\ell+1}^{*}\right.$
$\displaystyle\left.\hskip
80.0pt+\,\frac{(2\ell+1)}{\ell(\ell+1)}a_{\ell}b_{\ell}^{*}G_{\ell}G_{\ell}^{*}\right]$
(11)

Figure 3:  Axial force efficiency $Q_{z}$ as a function of position (in units
of sphere radius) along the laser axis for (a) sphere radius $r=1825\,{\rm
nm}$ with filling fractions $f=0$ (dotted line), $f=0.37$ (dashed line),
$f=0.5$ (solid line) and $f=0.55$ (solid-dotted line); and (b) $r=1500\,{\rm
nm}$ with $f=0$ (dotted line), $f=0.43$ (dashed line), $f=0.5$ (solid-dotted
line), and $f=0.58$ (solid line).

Most optical tweezers setups employ oil-immersion objectives. In this case,
the refractive index mismatch at the glass-water interface leads to spherical
aberration of the focused trapping beam, which can be taken into account by
introducing suitable phase factors Torok  in the expression (8) for the
multipole coefficients $G_{\ell}.$ The resulting Mie-Debye-spherical
aberration (MDSA) theory of optical tweezers Viana07  thus accounts for the
spherical aberration introduced by refraction at the glass slide. Excellent
blind agreement with experiments was obtained when additional optical
aberrations are also included Dutra12 ; Dutra14 .

##  III Results and Discussion

Figure 3a shows the optical axial force $Q_{z}$ variation with the microsphere
position $z$ with respect to the focal point (normalized by the radius
$r=1825\,{\rm nm}$). The parameters used in numerics are listed in Ref.
parameter . Figure 3a shows that an homogeneous ${\rm SiC}$ sphere ($f=0$) of
this size cannot be optically trapped since $Q_{z}>0$ for all values of $z.$
In fact, radiation pressure provides the dominant contribution to the optical
force due to the large refractive index of ${\rm SiC},$ thus leading to an
optical force pointing along the laser propagation direction for all values
for the sphere position. On the other hand, the presence of ${\rm SiO_{2}}$
inclusions allows for optical trapping since there are stable equilibrium
positions ($Q_{z}=0$) when considering $f=0.37$ and $f=0.5,$ corresponding to
the dashed and solid lines in Fig. 3a.

Figure 4:  (a) Scheme representing Mie scattering when the sphere center is
located at the focal point of the incident beam. In this case, the resulting
optical force $Q_{z}(z=0)$, plotted as a function of the sphere radius $r$ in
(b) for $f=0.5$ and in (c) for $f=0.58,$ can be interpreted in terms of the
interference between the fields reflected at planes $\Pi_{1}$ and $\Pi_{2}.$
(d,e) Backscattering efficiency $Q_{b}$ versus $r$ for (d) $f=0.5$ and (e)
$f=0.58$. The vertical lines indicate the filling fractions used in Figs. 5
and 6. Optimal trapping is achieved by selecting $r$ to be a simultaneous
minimum of both $Q_{z}$ and $Q_{b}$ (solid vertical lines).

The latter case corresponds to the situation where the first Kerker condition
$Q_{b}=0$ is fulfilled. Interestingly, this case corresponds precisely to the
situation where trapping is more stable, i.e. where the range of position
values for which the optical force is in opposition to the propagation
direction ($Q_{z}<0$) is larger, as indicated by the solid line in Figure 3a.
Besides, here it is important to emphasize that the effect of making the trap
more stable is not a trivial consequence of decreasing the effective
refractive index by increasing the relative amount of ${\rm SiO_{2}}$ in the
composite. Indeed, this can be seen by comparing the curves of $Q_{z}$ for two
slightly different values of the filling fraction, $f=0.5$ and $f=0.55$. While
the former leads to optimal optical trapping, in the latter case optical
trapping is impossible despite the fact that it corresponds to a larger
density of inclusions, and hence to a smaller effective refractive index as
shown in Fig. 1. This results unveils the crucial role played by minimizing
backscattering, achieved at the first Kerker condition as discussed in
connection with Fig. 2. We identify such configuration as providing the
optimal and more stable optical tweezing. All previous results are robust
against varying the host sphere radius, as illustrated in Fig. 3b, where an
analogous calculation is carried out considering $r=1500\,{\rm nm}.$ The first
Kerker condition is now satisfied at $f=0.58$, and the findings are
qualitatively the same, except that now there is a single equilibrium position
with an enlarged stability range along the $z$ axis, instead of two stable
equilibrium positions as in the case considered in Fig. 3a.

Figure 5: Axial force efficiency $Q_{z}$ as a function of position (in units
of sphere radius) along the laser axis for (a) $f=0.5$ with radii
$r=1300\,{\rm nm}$ (dashed line), $1630\,{\rm nm}$ (dotted line), and
$1825\,{\rm nm}$ (solid line); and for (b) $f=0.58$ with radii $r=1490\,{\rm
nm}$ (solid line), $1636\,{\rm nm}$ (dashed line), and $1700\,{\rm nm}$
(dotted line). Such radii correspond to minima of $Q_{z}(z=0),$ $Q_{b}$ or
both.  Figure 6: Same conventions as in Fig. 5, with the spherical aberration
introduced by the glass-water interface taken into account. Here $z=0$
represents the paraxial focal point, which is at a distance $L=7r$ above the
glass slide, as illustrated by the inset. The point of maximum energy density
(diffraction focus) is located in between the paraxial focus and the glass
slide ($z<0$).

In order to gain further insight into the conditions needed for optimal
trapping, we analyze the optical force $Q_{z}$ when the sphere center is
located at the focal point $(z=0).$ In this case, all optical rays forming the
incident laser beam propagate along radial directions and hence are not
deflected when refracting into the microsphere. In other words, all rays
composing the incident beam correspond to a vanishing impact parameter since
they intersect at the sphere center, as illustrated by Fig. 4a. We calculate
$Q_{z}(z=0)$ as a function of the sphere radius $r$ for $f=0.5$ (Fig. 4b) and
$f=0.58$ (Fig. 4c). Our results obtained within the wave-optical Mie-Debye
theory can be interpreted as resulting from the interference between the field
reflected at the water-composite interface (plane $\Pi_{1}$ in Fig. 4a) and
the one reflected at the composite-water interface (plane $\Pi_{2}$) after a
round-trip propagation along the sphere diameter epl . Indeed, the minima
shown in Figs. 4(b,c) are well approximated by

$r_{\rm min}=j\,\frac{\lambda_{0}}{4n_{\rm emg}},\,j\in\,\mbox{integers}$ (12)

with $\lambda_{0}/(4n_{\rm emg})\approx 131\,{\rm nm}$ ($f=0.50$) and
$136\,{\rm nm}$ ($f=0.58$), as in a slab of thickness $2r$ and refractive
index $n_{\rm emg}.$ When (12) is met, the two fields interfere destructively,
the reflectivity is minimized and so is the resulting radiation pressure, thus
favoring trapping.

The backscattering efficiency $Q_{b}$ for a plane wave cannot be interpreted
along the same lines. Within ray optics, an incident plane wave corresponds to
parallel optical rays which are deflected at the water-composite interface
along various angles depending on the impact parameter. Hence the variation of
$Q_{b}$ versus $r$ shown in Figs. 4d ($f=0.50$) and 4e ($f=0.58$) brings a
complementary information on trapping optimization. The first Kerker condition
corresponds to the radii giving a minimum $Q_{b}.$ The optimal trapping
conditions discussed in connection with Fig. 3a correspond to picking
$r=1825\,{\rm nm},$ which is a simultaneous minimum of both $Q_{z}(z=0)$ and
$Q_{b}$ as indicated by vertical solid lines in Figs. 4b and 4d. The
additional vertical lines at $r=1300\,{\rm nm}$ and $r=1630\,{\rm nm}$
indicate a minimum and a maximum of $Q_{z}(z=0)$ which turn out to be a
maximum and a minimum of $Q_{b},$ respectively. In Fig. 5a, we compare those
two configurations with the optimal one by plotting the corresponding
variations of the axial force versus position (in units of sphere radius). For
the former case (dashed line), radiation pressure is minimized near the focal
point, but then builds up as the sphere is displaced along the optical axis.
As a consequence, the resulting trapping range is significantly smaller. On
the other hand, for $r=1630\,{\rm nm}$ (dotted line), no trapping is possible
since radiation pressure is maximized near the focal point due to constructive
interference of the reflected field components shown in Fig. 4a. A similar
comparison is illustrated by Fig. 5b for the case $f=0.58,$ but now the
maximum of $Q_{z}(z=0)$ satisfying the first Kerker condition ($r=1700\,{\rm
nm}$, dotted line) does lead to trapping, although at a position far above the
focal point, and over a shorter stability range. Taken together, Figs. 3 and 5
show that selecting a microsphere satisfying the first Kerker condition
clearly helps to increase the trap stability. Nevertheless, the discussion
above indicates that the requirement (12) for destructive interference at the
focal point is more important for trapping high-refractive index microspheres
with the ideal, aplanatic (aberration-free) trapping beams considered so far.

The first Kerker condition becomes comparatively more important when spherical
aberration is taken into account. Indeed, the properties of optical tweezers
are extremely sensitive to small quantities of optical aberrations Dutra12 ;
Dutra14 . A very common source of spherical aberration is the refractive index
mismatch between the glass slide and water-filled sample when employing high-
NA oil-immersion objectives Torok . As discussed in the end of Sec. II.C, the
effect of the glass-water interface on the trapping beam is taken into account
within the Mie-Debye-spherical aberration (MDSA) theory of optical tweezers
Viana07 . In Fig. 6, we plot the MDSA results for the axial force efficiency
variation with the microsphere position, with the same parameters and
conventions employed in Fig. 5. We take the paraxial focus at a distance
$L=7\,r$, well above the glass slide, so that the optical reverberation
between the glass slide and the sphere is negligible Dutra2016 . As
illustrated in the inset of Fig. 6a, the energy density is distributed between
the paraxial focus ($z=0$) and the glass slide, and the diffraction focus is
located below the paraxial one. As a consequence, the stable equilibrium
positions shown in Fig. 6 are now located at $z<0.$ The comparison between
Figs 5 and 6 shows that the trap stability range is shorter and the maximum
restoring counter-propagating force is weaker, as expected since optical
aberrations degrade the focal region leading to a reduction of the energy
density gradients.

More importantly, Fig. 6 shows that our trapping proposal, with the radii and
filling fractions corresponding to the solid lines in Figs. 5 and 6, still
works even in the presence of a significant amount of spherical aberration.
The interpretation in terms of a parallel-planes interferometer (see Fig. 4a)
now holds only for the paraxial Fourier components of the incident beam, which
provide a significant fraction of the total radiation pressure. Thus,
selecting a radius satisfying the condition (12) for the ideal aplanatic case
is still helpful but no longer sufficient to achieve trapping, as illustrated
by the dashed lines in Fig. 6. In fact, trapping is now possible only by
picking a radius from (12) that also turns out to minimize $Q_{b},$ as
indicated by the solid vertical lines in Fig. 4.

##  IV Conclusion

Recent advances in the field of metamaterials have made possible the
achievement of highly directional Mie scattering in the optical domain. We
have shown that the condition of zero backward scattering (first Kerker
condition) can be applied to optically trap high-index microspheres with a
single focused laser beam, thereby widening the range of applications of this
popular experimental technique.

We have considered SiC microspheres with ${\rm SiO_{2}}$ inclusions immersed
in water. We have shown that backward scattering can be strongly suppressed by
tuning the filling fraction. In such cases, stable trapping is possible
provided that the sphere radius and filling fraction are such as to also lead
to a destructive interference between the fields reflected at the external and
internal interfaces illustrated by Fig. 4a.

Fulfilling the first Kerker condition considerably enhances stability, but is
not a necessary condition for trapping when optical aberrations are
disregarded. On the other hand, for typical setups with oil-immersion high-NA
objectives, it is essential to satisfy the Kerker condition in order to
achieve trapping of high-index microspheres, for the refractive index mismatch
between the glass slide and the water-filled sample introduces a significant
amount of spherical aberration.

###### Acknowledgements.

We thank D. S. Ether jr and N. B. Viana for inspiring discussions. This work
has been supported by the Brazilian agencies National Council for Scientific
and Technological Development (CNPq), the National Institute of Science and
Technology Complex Fluids (INCT-FCx), the Carlos Chagas Filho Foundation for
Research Support of Rio de Janeiro (FAPERJ) and the São Paulo Research
Foundation (FAPESP). F. A. P. thanks the Royal Society – Newton Advanced
Fellowship (grant no. NA150208) for financial support.

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