Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17463

Super-resolution fluorescence imaging ofcarbon nanotubes using a nonlinear excitonic
process
K EIGO OTSUKA , 1,2 A KIHIRO I SHII , 1,2 AND Y UICHIRO K. K ATO 1,2,*
1 Quantum Optoelectronics Research Team, RIKEN Center for Advanced Photonics, Saitama 351-0198,

Japan
2 Nanoscale Quantum Photonics Laboratory, RIKEN Cluster for Pioneering Research, Saitama 351-0198,
Japan* yuichiro.kato@riken.jp

Abstract: Highly efficient exciton-exciton annihilation process unique to one-dimensional
systems is utilized for super-resolution imaging of air-suspended carbon nanotubes. Through thecomparison of fluorescence signals in linear and sublinear regimes at different excitation powers,
we extract the efficiency of the annihilation processes using conventional confocal microscopy.
Spatial images of the annihilation rate of the excitons have resolution beyond the diffraction
limit. We investigate excitation power dependence of the annihilation processes by experimentand Monte Carlo simulation, and the resolution improvement of the annihilation images can bequantitatively explained by the superlinearity of the annihilation process. We have also developed
another method in which the cubic dependence of the annihilation rate on exciton density isutilized to achieve further sharpening of single nanotube images.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1.

Introduction

As a result of strong Coulomb interaction arising from the one-dimensional (1D) nature of
single-walled carbon nanotubes (CNTs), electron-hole pairs form excitons that are stable evenat room temperature [1–3]. When multiple excitons exist in a CNT, an exciton can scatterwith another exciton and decay nonradiatively. In this Auger process known as exciton-exciton
annihilation (EEA) [4–6], the exciton recombines by giving its energy to the other exciton.
Confinement and diffusion [7–10] of the excitons in a nanotube lead to efficient EEA process
upon collision with one another, resulting in a peculiar cubic dependence of the EEA rate on thedensity of excitons [11, 12]. The efficient EEA can be, for example, utilized in room-temperaturesingle photon generation at telecommunication wavelengths [13, 14]. The diameter-dependent
wavelength of nanotube fluorescence also includes the near-infrared window, where scattering
is small and absorption is weak, allowing for deep-tissue imaging using CNTs as fluorescentagents [15–18]. As advanced techniques of super-resolution imaging [19–22], such as two-photon
excitation microscopy and stimulated emission depletion microscopy, rely on the nonlinear
optical response in fluorescence agents [23–28], the EEA process could play a key role in thedevelopment of nanotube-based biological imaging as well.
Here we demonstrate subdiffraction imaging of air-suspended CNTs by extracting the nonlinear
EEA component using a typical confocal microscopy system. By combining two fluorescence
images obtained at different excitation powers, an EEA rate image with enhanced resolution canbe constructed. Excitation power-dependence of the extracted EEA efficiency and the spatial
resolution of the EEA imaging are experimentally investigated, and we perform Monte Carlosimulation of the EEA process to identify the resolution limit of this technique. In addition to theuse of nonlinearity between the EEA rate and the exciton generation rate, the cubic dependenceof the EEA rate on exciton density is utilized in another protocol for super-resolution imaging of#363745
Journal © 2019https://doi.org/10.1364/OE.27.017463
Received 29 Mar 2019; revised 17 May 2019; accepted 22 May 2019; published 10 Jun 2019

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17464

CNTs. By measuring the excitation power required to establish a predefined exciton density, we
are able to achieve even narrower widths for isolated nanotube images.
2.

Air-suspended nanotubes and optical setup

The air-suspended nanotubes are synthesized by alcohol chemical vapor deposition [11, 29].
Trenches are formed on Si substrates through electron beam lithography and dry etching. We use
Fe(III) acetylacetonate and fumed silica in ethanol as catalyst, where spin-coating and lift-off
processes are used to deposit the catalyst in a region defined by another lithography process.
After heating in air at 400◦ C for 5 min, CNTs are synthesized at 800◦ C for 1 min using Ar andH2 flowing through an ethanol bubbler. A schematic and a scanning electron micrograph of an
air-suspended nanotube are shown in Figs. 1(a) and 1(b), respectively.
A homebuilt confocal microscopy system is used to perform photoluminescence (PL) measurements at room temperature [11, 30–32]. A wavelength-tunable Ti:sapphire laser is used for
excitation after controlling its power and polarization by neutral density filters and a half-wave
plate, respectively. The laser beam is focused on the samples using an objective lens with anumerical aperture of 0.8 and a working distance of 3.4 mm. PL is collected through the same
objective lens and detected using a liquid-nitrogen-cooled InGaAs diode array attached to aspectrometer. With the laser beam position fixed, we scan the samples mounted on a motorized
three-dimensional stage to achieve focusing, sample search over the entire chip, and imagingof the single tubes. All measurements are performed in dry nitrogen to avoid formation ofoxygen-induced defects [9, 33].
CNT

(a)

(b)

CNT

(d)

(c)
850

800

300
150

IPL (104 counts/s)

Catalyst

900

PL (counts/s)

Excitation length (nm)

Laser beam

1
0.5
0

0

750
1200
1400
1600
Emission wavelength (nm)Fig. 1. (a) A schematic of an air-suspended nanotube sample. For optical imaging,
the samples are scanned along trenches relative to the fixed laser beam. (b) A scanningelectron micrograph of a typical nanotube. (c) A PL excitation map for a (11,3) nanotube.
The excitation power is 0.1 µW. (d) A PL image of the nanotube measured in (c). Inset:
Polarization dependence of IPL . Excitation power and wavelength used for imaging are
0.1 µW and 780 nm, respectively, and the image is extracted at an emission wavelength of
1175 nm with a spectral integration window of ∼50 nm. Scale bars in (b) and (d) are 500 nm.

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17465

3.

Extraction of the influence of exciton-exciton annihilation

0.9

100
10−1
10−2
10−2 10−1 100 101 102
Excitation power (μW)

0.7

0.5

Normalized
intensity (a.u.)(a)

FWHM (μm)

IPL (a.u.)101

IPL (a.u.)

Figure 1(c) shows PL excitation spectroscopy data performed on a single air-suspended nanotube
with a length L ≈ 1.2 µm, and we determine the tube chirality to be (11,3). We then investigate
laser polarization dependence of PL intensity with excitation energy at the E22 resonance [inset
of Fig. 1(d)]. For the following PL measurements, the polarization angle and the excitation
wavelength are fixed parallel to the tube axis and at the E22 resonance, respectively. Figure 1(d)shows an image of the integrated PL intensity IPL obtained with a ∼50-nm-wide spectral
integration window centered at the E11 resonance. The width of the nanotube PL image ispredominantly determined by the excitation laser beam profile, and thus much larger than theactual nanotube diameter of 1.0 nm.
The resolution enhancement will be achieved by using the efficient EEA process, whose
effects can be observed in the power dependence of PL intensity [Fig. 2(a), black squares]. IPL
is proportional to the excitation power at low powers, but shows a cubic root dependence athigh powers due to EEA [11]. The blue line in Fig. 2(a) is a linear fit to the low-power results,
corresponding to the PL intensity expected in the absence of EEA. The deviation of actual PL
intensity from the blue line would correspond to the EEA rate ΓEEA .
The efficient EEA process also affects the imaging resolution. We measure 1D IPL profilesof the nanotube along the trench direction with various excitation powers, and the full-width at
half-maximum (FWHM) wh of the profiles are plotted as red circles in Fig. 2(a) as a function of
the excitation power. At the low power limit, PL intensity profiles reflect the excitation laser beamprofile because IPL is proportional to the exciton generation rate g in the absence of EEA. At theother extreme where IPL is proportional to g 1/3 due to the cubic-law EEA process, we expect theeffects of nonlinearity on the width of the PL intensity profile. If we approximate the laser beamprofile by a Gaussian function exp(−2x 2 /r 2 ) with r being the laser 1/e2 radius, the intensity√
profile of PL that has g α dependence becomes IPL (x) ∝ exp(−2αx 2 /r 2 ) = exp[−2x 2 /(r/ α)2 ],where α is the power exponent of the generation rate dependence. The FWHM changes by afactor√ √r/ α 2 ln 2
1
=√ ,
(1)
√
α
r 2 ln 2√
and it is reasonable that the width of IPL profile increases by a factor of 3 at the highest power.

4

(b)
x13

2
0
1

0

0.2 μW
2.6 μW(c)−1

0x position (μm)

0.2 μW
2.6 μW
EEA

1Fig. 2. (a) Excitation power dependence of PL intensity IPL with the laser focused at
the center of the nanotube (black squares) and FWHM of PL intensity profiles along thetrench direction (red circles). The nanotube is the same one as in Fig. 1(c) and 1(d). IPL
is obtained by calculating the peak area of a Lorentzian fit to the emission spectrum. (b)PL intensity profiles scaled by excitation powers. Difference of the black and the blue lines
is proportional to the EEA rate ΓEEA . (c) Normalized profiles of IPL for the two different
excitation powers and the EEA rate obtained from the subtraction in (b).

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17466

Although it may seem as if the EEA process has a negative effect on resolution, we can utilize
the nonlinear power dependence on the generation rate to achieve enhanced spatial resolution.
We extract the EEA component from PL intensities measured at two different excitation powers,
where the EEA extraction power P1 is higher than the reference power P2 = P1 /κ, where κ > 1.
Figure 2(b) illustrates how the EEA component is extracted. The black line is the PL intensity
profile IPL (x, P1 = 2.6 µW), showing slight broadening caused by EEA. The blue line is theintensity profile IPL (x, P2 = 0.2 µW) scaled by κ, which is equivalent to the expected PL intensity
profile in the absence of the EEA process. The difference between the black and blue lines
κIPL (x, P2 ) − IPL (x, P1 ) (red region) is proportional to the EEA rate ΓEEA in the nanotube.
Figure 2(c) shows normalized profiles of raw PL intensity (black and blue for P1 and P2 ,respectively) and the EEA component extracted by the protocol described above (red). The
width of the intensity profile at the excitation power of 0.2 µW corresponds to the minimum
width obtained in the power-dependent width in Fig. 2(a), but the profile of the extracted EEA
component has an even smaller width. As ΓEEA increases superlinearly with g, the spatial profileof the EEA component peaks sharply when the laser is centered on the nanotube.
To characterize the resolution and signal-to-noise ratio, we perform experiments using a rangeof excitation power combinations for extracting the EEA rate ΓEEA . Measurements are repeated
7 times for each combination of the EEA extraction power P1 and the reference power P2 , andthe average width of the ΓEEA profiles is plotted in Fig. 3(a) as a function of P1 and P2 . Roughly
speaking, large P1 and P2 result in large wh of the ΓEEA profiles, while low powers are preferable
for high-resolution imaging.
We consider the widths obtained at the lowest P2 = 0.05 µW to quantitatively evaluate theresolution improvement. Figure 3(b) shows the P1 dependence of wh obtained from the raw PL
intensity profiles (black squares) and the extracted EEA profiles with P2 fixed at 0.05 µW (redcircles). Error bars represent the standard deviation for the repeated measurements. The FWHM
from raw PL reproduces the behavior observed in Fig. 2(a), and the error values are similar overthe entire range of the excitation power. In comparison, the width for EEA rate profiles decreases
from∼500 to ∼350 nm as P1 decreases, which is approximately an improvement by a factor of√
2. We note that the width of the EEA profiles has large error bars at low powers because of thesmall ΓEEA signals that depends superlinearly on g, becoming comparable to photon shot noiseand detector noise.
4.

Comparison with Monte Carlo simulation results

The minimum width achievable from the EEA profile in an ideal situation with negligible noise isinvestigated by conducting Monte Carlo simulation [11, 34] using parameters directly comparablewith the experimental results. We use the same method as [11], and modify the excitation profile
to reproduce the spatial scanning of the sample. Excitonic processes, such as exciton generation,diffusion, and decay, are evaluated at time intervals ∆t ≤ 10−3 τ, where τ is the lifetime of theexciton intrinsic lifetime. For exciton generation, photons are supplied into the system at a
rate with a spatial probability distribution given by a Gaussian function exp[−2(x 2 + y 2 )/r 2 ],where x and y are positions perpendicular to and along the nanotube, respectively. Excitons are
generated from the photons on the nanotube with a photon absorption width of 5 nm. Note that awidth thicker than the physical width of the nanotube does not affect the following discussion
based on the number of generated excitons gτ, and it is used to reduce computational load.
Probability for the exciton displacement s due to diffusion is given by the normal distributions2√ 1
exp(− 4D∆t
) with D being the diffusion constant. In addition to the intrinsic decay process
4πD∆tthat occurs with the probability of ∆t/τ, the excitons go through extrinsic decay (end quenchingand EEA) after the diffusion. The tube√length L and the laser radius r are 1.2 µm and 0.4 µm,
respectively, while the diffusion length Dτ = 1 µm is assumed.

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17467

gτ

Intrinsic decay

(b)

102

Excitation power (μW)
EEA
10−2

(d)
0.7

0.5

105

10−1

100
101
P1 (μW)N

10−1 100

101

(f)

104
103

0.5

101
100

101
100

10−1

100

101
gτ

102

104
103
102

102

0.3

0.3

Γτ

101

100

FWHM (μm)

FWHM (μm)

0.7

100

0.3

ΓEEA (a.u.)

Raw PL100
101
P1 (μW)
EEA (P2=0.05 μW)

101

0.5

ΓEEAτ

0.3

0.7

10−2 10−1 100 101 102 103
104
(e)
102
103
experiment
101
EEA
102
100
101
PL
10−1
100
simulation 10−2
10−1
10−2 10−1 100 101 102

Intensity (a.u.)

0.5

10−1
10−1

102

FWHM (μm)

0.7
gτ

100

FWHM (μm)

P2 (μW)

101

gτ

(c)

(a)

10−1 10010−1experiment −2
simulation 10
10−3
1
10
102 103

PL intensity (a.u.)

Fig. 3. (a) FWHM of 1D EEA profiles as a function of the EEA extraction power P1 andthe reference power P2 used for the subtraction of IPL . The data are averaged for sevenmeasurements repeated with the same condition. (b) FWHM of IPL and the extracted EEA
rate profiles for a fixed power of P2 = 0.05 µW. Error bars represent the standard deviation.
√
Solid and broken gray lines correspond to the laser beam FWHM and that multiplied by 1/ 2,
respectively. (c) FWHM as a function of the EEA extraction and the reference generation
rate from the Monte Carlo simulations. (d) FWHM of the intrinsic decay rate profiles and the
EEA rate profiles from the simulations. (e) Excitation power dependence of the PL intensity
and the extracted EEA component for the experiments and the simulations. (f) EEA rates asa function of IPL and N from the experiments and the simulations, respectively.

1D rate profiles are calculated by moving the nanotube position in the x direction, simulating
the experiment performed to obtain the IPL profiles shown in Fig. 2(b). We compute the intrinsic
decay rate ΓI from the time-averaged number of excitons that go through the intrinsic decay,
which corresponds to the PL intensity in the experiments. The simulations are repeated forvarious photon incident rates, and the EEA rate profiles are computed from ΓI profiles in amanner similar to the measurements for Fig. 3(a). Figure 3(c) shows the FWHM of the extracted
EEA rate profiles as a function of the EEA extraction and reference generation rates. Data areplotted in terms of a unitless parameter gτ, representing the average number of excitons that are
generated during τ. Note that we use the exciton generation rate evaluated with the nanotube at
the center of the laser beam in Fig. 3, in which case it becomes equivalent to the excitation powerin the experiments. The simulation results exhibit a trend similar to the experiments; large wh for
combinations of large gτ and vice versa.
Using the simulation data, it is possible to directly evaluate the EEA rates ΓEEA . In Fig. 3(d),the FWHM of the EEA profiles as well as the intrinsic decay profiles are plotted, and both
reproduce the overall behavior of the experimental data. As the generation rate is decreased, thewidth of the intrinsic decay approaches that of the simulated laser profile as indicated by thesolid gray line. The EEA profile width, in comparison, is already below the solid line at large
generation rates. The width √decreases with the generation rate, and approaches the broken grayline which corresponds to 1/ 2 width of the laser profile. According to Eq. (1), the change in whVol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17468

implies that the EEA rate has a nontrivial dependence on the exciton generation rate.
We experimentally confirm the g dependence of ΓEEA by measuring the PL intensity [Fig. 3(e),open circles] and subtracting a linear fit to the low power PL data (open diamonds). Similarly,
the gτ dependence of the intrinsic decay rate (filled circles) and the EEA rate (filled diamonds)
obtained from the simulation is overlaid in Fig. 3(e) in terms of unitless parameters ΓI τ andΓEEA τ, respectively. The simulation is in good agreement with the experiment, showing that oursimple model accurately describes the behavior of excitons in nanotubes. In this log-log plot,ΓI has slopes of 1 and 1/3 at low and high generation rates, respectively, as also observed inFig. 2(a). The slope of ΓEEA is of our interest, which is 2 at low powers and approaches 1 as thepower is increased. Through Eq. (1), the observed values of the slope can quantitatively explain
the power dependence of wh obtained from the EEA profiles [Fig. 3(d)].
The ΓEEA slope of 1 at high power is reasonable, because g ≈ ΓEEA with EEA being thedominant decay process. The quadratic dependence of ΓEEA , however, seems to contradict withthe cubic dependence expected for EEA [11]. In order to resolve this apparent inconsistency,
ΓEEA τ (filled diamonds) is plotted as a function of time-averaged exciton number N = ΓI τ
[Fig. 3(f)], and we find a transition from the quadratic to cubic dependence near N = 1. Similarly,
the experimentally extracted EEA rate is replotted as a function of the PL intensity (open
diamonds) in Fig. 3(f), which coincides with the simulation result. The transition in the N
dependence of the EEA rate can be explained by the fact that EEA process occurs only when
multiple excitons coexist in a nanotube. When N > 1 and the inter-exciton distance is small, theEEA rate in a 1D system is proportional to N 3 as discussed in [11]. When the exciton numberN  1, however, the situation changes. The probability for the instantaneous exciton number
λ ≥ 3 is negligible, while EEA does not occur for λ = 0 and 1. The EEA rate is then dominatedby the case where two excitons coexist, whose probability is given by the Poisson distributionp(λ) = N λ e−N /λ! to be p(2) ≈ N 2 /2. In this regime, multiple exciton generation, rather than
exciton diffusion, is the limiting factor of the EEA process.
We can further obtain an explicit expression for the EEA rate. After time t from the firstexciton generation, the survival probability of the first exciton and the arrival probability of thesecond exciton are ps (t) = exp(−t/τ) and pa (t) = gt exp(−gt), respectively. The expectation
value of the arrival time interval for when two excitons coexist is∫∞t ps (t)pa (t)dt
2τ
∫0 ∞
=
≈ 2τ,
(2)
gτ
+1
ps (t)pa (t)dt
0being independent of the exciton generation rate g as longpas N  1. The initial distance between
the two excitons after diffusion is therefore constant (∼ 8Dτ/π). Despite the diffusion of thefirst exciton, the collision rate of such exciton pairs is then ∼ π/2τ under the condition of λ = 2,
and the EEA rate is given byπ
πN 2ΓEEA ≈ p(2) ×
=
.
(3)
2τ
4τ
5.

Two-dimensional images of exciton-exciton annihilation rates

Having understood the mechanism for the resolution improvement, we perform two-dimensional(2D) imaging of the raw PL intensity and the EEA rate. According to the analysis performed in
Figs. 3(a) and 3(c), the reference power P2 should be small enough to avoid EEA while the EEA
extraction power P1 also should be in the linear regime of IPL vs the excitation power in order toobtain high resolution images. We note, however, that P1 should not be too close to P2 to keep
the signal-to-noise ratio sufficiently high. We thus choose P1 = 0.8 µW and P2 = 0.1 µW, where
the excitation power is modulated by switching the neutral density filters at every step of thesample scan.

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17469

(d)

(g)
IPL (a.u.)

(a)

1
0.5
0

(e)

(h)

Intensity (a.u.)

ΓEEA (a.u.)

(b)

(c)

(9,7)

(f)

1
0.5
0

(i)
(9,7)

(9,8)

Fig. 4. 2D images of a (9,7) nanotube for (a) the PL intensity and (b) the extracted EEA rate.
(c) 1D profiles from the PL (black) and EEA (red) images at the same position indicated bybroken white lines. (d–i) Similar sets of 2D images and the 1D profiles for the two nanotubes
with (d–f) a parallel and (g–i) a Y-shaped configuration. The excitation wavelengths are
fixed at the E22 (784, 782, and 802 nm for (a–c), (d–f), and (g–i), respectively) of eachnanotube, and all the images are extracted at the E11 wavelength (1284, 1290, and 1373 nm
for (a–c), (d–f), and (g–i), respectively) with a spectral window of ∼50 nm. All the images arenormalized by their maximum intensity so that the image resolution can be easily compared.
Scale bars are 500 nm.

Figure 4 displays the optical images for three configurations of nanotubes. Because P2 = 0.1 µW
used for Figs. 4(a), 4(d), and 4(g) is in the linear regime of IPL , the width of the raw PL image issolely limited by the laser beam profile and cannot be further reduced by lowering the power. In
the simplest case of a single nanotube, the EEA image of the (9,7) nanotube [Fig. 4(b)] has evensmaller width than the PL image [Fig. 4(a)]. We note that the EEA image represents the degreeof PL intensity reduction through the EEA process. When two (9,7) tubes are lying closely, thetubes are more clearly resolved by extracting the EEA component, which is also apparent fromthe 1D profiles shown in Fig. 4(f). For a Y-shaped junction of (9,8) nanotubes, we show 1D
profiles at the position where the IPL profile of the two tubes cannot be resolved [black line in
Fig. 4(i)]. These nanotubes are successfully separated through the super-resolution imaging of
EEA rates.
The EEA imaging technique may potentially be applied to three-dimensional biologicalimaging. We expect similar resolution enhancement in the depth direction as the laser intensity
profile along this axis can also be approximated by a Gaussian function. It is also worth
mentioning that the nanotube lengths in the EEA images are slightly shortened because the
EEA rate is small near the nanotube ends where the exciton density is low. In principle, the
FWHM along the tube axis could become smaller than the diffraction limit by using short CNTs,
though the resolution improvement would be quantitatively different from the other axes. It isnoteworthy that the EEA-derived nonlinearity in nanotubes enables subdiffraction imaging at a
power density as low as ∼300 W/cm2 with a continuous wave laser, whereas considerably higher
intensities are employed for other subdiffraction techniques that utilize similar algorithms relying
on PL saturation in dyes [26, 28]. We also note that this method is compatible with dispersed
nanotubes used for biological imaging [15, 16, 18]. No modification of the protocol should beVol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17470

needed, but the useful power range for EEA imaging will depend on the nanotube length andexciton diffusivity.
6.

Image sharpening using the cubic-law excitonic process at high powers

We now consider whether it is possible to utilize the cubic dependence of the EEA rate ΓEEA on
N to achieve even higher resolution images. In the imaging protocol employed in Figs. 2–4, theg 2 dependence of ΓEEA is used [Protocol I, left panel of Fig. 5(a)]. In comparison, we introduce a
protocol that exploits the N 3 term of the EEA rate that appears at large N [Protocol II, right panelof Fig. 5(a)]. Instead of laser position dependent exciton generation rate g(x) resulting from a
constant excitation power in Protocol I, we create laser position dependent exciton number N(x)that reproduces the laser beam profile. The EEA rate extracted from such N(x) should allow
for resolution enhancement through the cubic dependence. As briefly illustrated in Fig. 5(b),we first obtain a PL intensity profile I0 (x) at a sufficiently low excitation power P0 to purely
extract the generation rate profile g0 (x). Then I0 (x) is multiplied by a constant κ > 1, which
will be the target PL intensity κI0 (x) for the next scan. We reproduce the target PL intensity
profile κI0 (x) by controlling the excitation power P(x) to compensate for the intensity loss due
to EEA. P(x) is tuned at every point until the intensity difference between the measurementand κI0 (x) becomes less than 3%. The position-dependent exciton generation rate for thisscan g(x) = g0 (x)P(x)/P0 is used to extract the EEA rate ΓEEA (x). As the EEA rate is equalProt. I

Prot. II

κI0(x)

𝜅(ii) Power tuning to get κI0(x)∝N2Generation rate

κI0(x)∝ N(x)(iii) Conversion to EEA rate

ΓEEA (a.u.)g = g0P/P0

1
0.5

1
0.5
0

(e)
Normalized intensity

IPL (a.u.)

Exciton number0(d)

P0: const.
I0(x) ∝ g0(x)

P(x)

∝g2

(c)

(b) (i) Low power measurement

∝N3

ΓEEA

∝g

ΓEEA

(a)

Δg = g–κg0 = ΓEEA

(f)
PL
EEA

−1 −0.5 0 0.5 1
x position (μm)

PL
EEA

−1 −0.5 0 0.5 1
x position (μm)

Fig. 5. (a) Illustration for utilizing the nonlinearity of ΓEEA as a function of the exciton
generation rate g or the exciton number N. While quadratic nonlinearity of ΓEEA at small gis used for Protocol I (used in Figs. 2–4), cubic nonlinearity of ΓEEA against N is used inProtocol II. (b) Schematics of the protocol to extract the cubic nonlinearity. (c) PL and (d)EEA images of the (11,3) nanotube with P0 = 0.2 µW and κ = 25. Excitation and extraction
wavelengths are 781 and 1176 nm, respectively. Scale bars are 500 nm. (e) 1D intensity
profiles of the nanotube shown in (c) and (d). (f) Intensity profiles of the nanotubes shownin Figs. 4(d)–4(f). Excitation and extraction wavelengths are 780 and 1290 nm, respectively.

Vol. 27, No. 13 | 24 Jun 2019 | OPTICS EXPRESS 17471

to the additional exciton generation rate compared with that expected in the absence of EEA,ΓEEA (x) = ∆g(x) = g(x) − κg0 (x).
Figures 5(c) and 5(d) show the raw PL image and the EEA rate image, respectively, obtained
through Protocol II. Since larger κ gives smaller wh , we choose κ = 25 and P0 = 0.2 µW to
fully utilize the N 3 nonlinearity at high powers, resulting in the maximum P(x) = 53 µW. Cross
sectional profiles of these images are shown in Fig. 5(e). Clearly, the EEA rate image shows anarrower profile. If we take horizontal cross sections of the images and average the width along
the length of the tube, the EEA image gives an average wh of 290√ nm, while that for the PL3
image is 520 nm. The width reduction corresponds to a factor of 3 as expected from the N
dependence. As shown in Fig. 5(f), this method can resolve multiple CNTs lying closely as aresult of the improved spatial resolution. It should be noted, however, that the spatial resolutionof the image does not always improve as much as the width observed for single tubes. Unlike thecase in Protocol I, where the linear component of the PL is canceled out during the subtraction,
the linear component from adjacent nanotubes is included in the signal during the power tuning
step of Protocol II. The inclusion of the linear component inevitably causes a reduction of thenonlinear EEA component, resulting in a suboptimal resolution for high excitation powers.
7.

Conclusion

In summary, we have performed super-resolution imaging of CNTs by visualizing the efficient
EEA process using two protocols compatible with typical confocal microscopy systems. By
subtracting the linear PL√component to extract the nonlinear EEA rate, the spatial resolution
improves by a factor of 2 compared to the diffraction limit. The resolution improvement isdetermined by the N 2 dependent EEA rate at low exciton numbers, as confirmed by Monte
Carlo simulations. To utilize the N 3 dependence unique to the 1D system of nanotubes, wehave developed another protocol for super-resolution imaging in which the excitation power isadaptively tuned during the measurement.
Using the second protocol, the width of a single√
nanotube can be as narrow as 1/ 3 compared to the conventional PL imaging. The absorption
and emission at the near-infrared window and the nonlinear response that appears at low excitation
density would open up a pathway for the resolution enhancement in applications such as deep
tissue imaging with suppressed damages.
Funding
JSPS (KAKENHI JP16H05962 and JP17H07359); RIKEN (Incentive Research Project); MEXT
(Nanotechnology Platform).
Acknowledgments
We thank the Advanced Manufacturing Support Team at RIKEN for technical assistance.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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