Supplementary Information
Supercontinuum spatial modulation spectroscopy: Detection and noise limitations
M. P. McDonald,†,11 F. Vietmeyer,†,1 D. Aleksiuk,2 and M. Kuno*,12
1. Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame,
IN, 46556, USA
2. Department of Physics, Taras Shevchenko National University of Kiev, Kiev, 01601,
Ukraine
I. Calculating extinction cross section (𝝈𝒆𝒙𝒕 ) from lockin voltage.
Assuming a laser’s intensity profile at the beam waist is TEM00, its electric field [𝐸(𝑥)] can be
approximated as a Gaussian distribution, where
𝐸(𝑥) = 𝐸0 Exp [−
(𝑥 − 𝑥0 )2
].
2𝑠 2
(1)
Since intensity is the square of the electric field [𝐼(𝑥) = |𝐸(𝑥)|2 ], Equation 1 becomes
𝐼(𝑥) = 𝐼0 Exp [−
[(𝑥 − 𝑥0 )2 ]
]
𝑠2
(2)
where 𝐼0 is the maximum intensity at position 𝑥0 , and s is the standard deviation. For nominal values of
𝐼0 = 1, and 𝑥0 = 0, the point where the intensity falls to 𝐼0 /𝑒 2 (given as 𝑥 = 𝜔0 ) is found by
†
*
M. P. McDonald and F. Vietmeyer contributed equally to this work.
Author to whom correspondence should be addressed. E-mail: mkuno@nd.edu
1
(𝜔0 )2
1
1
𝜔02
=
Exp
[−
]
→
−ln
(
)
=
→ 2𝑠 2 = 𝜔02 → 𝜔0 = 𝑠√2 .
𝑒2
𝑠2
𝑒2
𝑠2
(3)
Using Equations 2 and 3, the Gaussian profile is described in terms of the “1/𝑒 2 radius”, giving
𝐼(𝑥) = 𝐼0 Exp −
[
(𝑥 − 𝑥0 )2
𝜔 2
( 0)
]
√2
= 𝐼0 Exp [−
2(𝑥 − 𝑥0 )2
].
𝜔0 2
(4)
Figure S1: Gaussian intensity distribution showing the relationship between standard deviation (𝑠) and
the 1/𝑒 2 radius of the beam (𝜔0 ).
The cross-sectional intensity profile of a TEM00 beam is approximated as a 2 dimensional
Gaussian profile, giving
𝐼(𝑥, 𝑦) = 𝐼0 Exp [−
2[(𝑥 − 𝑥0 )2 + (𝑦 − 𝑦0 )2 ]
].
𝜔02
(5)
The factor 𝐼0 is determined by normalizing Equation 5 to the incident power (𝑃0 ) with 𝑥0 = 𝑦0 = 0 for
simplicity.
2
∞
∞
∫ ∫ 𝐼0 Exp (−
−∞ −∞
2𝑥 2
2𝑦 2
Exp
)
(−
) 𝑑𝑥𝑑𝑦 = 𝑃0
𝜔02
𝜔02
(6)
2
2
2𝑃0
𝐼0 = 𝑃0 √ 2 √ 2 =
𝜋𝜔0 𝜋𝜔0 𝜋𝜔02
When an absorbing entity is moved into the beam waist, it extinguishes some of the intensity by
both absorbing and scattering the incident radiation. The extent of extinction is determined by an
extinction cross section (𝜎𝑒𝑥𝑡 ). However, since the beam’s distribution is non-uniform, the analyte’s
position must be defined as (𝑥𝑃 , 𝑦𝑝 ) = (𝑥 − 𝑥0 , 𝑦 − 𝑦0 ) to determine how much incident intensity it is
being subjected to. In addition, SCSMS works by shaking the analyte sinusoidally at a specific frequency
(𝑓) and direction (𝑥, in our case). Finally, setting 𝑦𝑝 = 0, the analyte’s position relative to the Gaussian
beam is given by 𝑥𝑝 = (𝑥 − 𝑥0 ) + 𝛿 sin(2𝜋𝑓𝑡) where 𝛿 is half the modulation amplitude (Figure S2).
Setting 𝑥0 = 0, the extinguished power is 𝑃 = 𝜎𝑒𝑥𝑡 𝐼(𝑥, 𝑡), giving a transmitted power of
𝑃𝑡𝑟𝑎𝑛𝑠 (𝑥, 𝑡) = 𝑃0 − 𝜎𝑒𝑥𝑡 𝐼(𝑥, 𝑡)
or
𝑃𝑡𝑟𝑎𝑛𝑠 (𝑥, 𝑡) = 𝑃0 − 𝜎𝑒𝑥𝑡
2𝑃0
2[(𝑥 + 𝛿 sin(2𝜋𝑓𝑡))2 ]
Exp
(−
).
𝜋𝜔02
𝜔02
(7)
(8)
A photodetector exposed to this power will give a voltage
𝑉𝑑𝑒𝑡 (𝑥, 𝑡) = 𝑅𝐺 [𝑃0 − 𝜎𝑒𝑥𝑡
2𝑃0
2[(𝑥 + 𝛿 sin(2𝜋𝑓𝑡))2 ]
Exp
(−
)]
𝜋𝜔02
𝜔02
3
(9)
where R is the responsivity of the photodiode (A/W) and G is the transimpedance gain (V/A). Here, f is
determined by the experiment (750 Hz) and  is measured by imaging a modulated fluorescent
polystyrene bead with a CCD camera (Figure S2).
Figure S2: Emission profile of a 200 nm polystyrene fluorescent bead being modulated at 750 Hz. The
piezo driver is being fed a 750 Hz sine wave (x is the modulation direction). The total distance (2) is
the peak-to-peak distance, where xc is the center position of the sphere.  is found to be 386 nm.
The waveform in Equation 9 can be expanded with a Fourier series. Specifically, if
𝑓(𝑥𝑝 ) = Exp (−
2(𝑥𝑝 )2
)
𝜔02
(10)
Then the Fourier series expansion of 𝑓(𝑥𝑝 ) is defined as
∞
𝑎0
𝑓𝐹 (𝑥𝑝 ) =
+ ∑[𝑎𝑛 cos(𝑛2𝜋𝑓𝑡) + 𝑏𝑛 sin(𝑛2𝜋𝑓𝑡)]
2
𝑛=1
4
(11)
where 𝑓 is the frequency and
1
2𝑓
𝑎0 = 2𝑓 ∫
−
1
2𝑓
𝑎𝑛 = 2𝑓 ∫
−
1
2𝑓
1
2𝑓
𝑏𝑛 = 2𝑓 ∫
−
1
2𝑓
1
2𝑓
𝑓(𝑥𝑝 )𝑑𝑡
𝑓(𝑥𝑝 ) cos(𝑛2𝜋𝑓𝑡) 𝑑𝑡
(12)
𝑓(𝑥𝑝 ) sin(𝑛2𝜋𝑓𝑡) 𝑑𝑡
Odd values of n result in 𝑎𝑛 = 0 and even values of n give 𝑏𝑛 = 0. From this, Equation 9 can be
written as
𝑉𝑑𝑒𝑡 (𝑥, 𝑡) = 𝑅𝐺𝑃0 [1 − 𝜎𝑒𝑥𝑡
2 𝑎0
( + 𝑏1 sin(2𝜋𝑓𝑡) + 𝑎2 cos(4𝜋𝑓𝑡) + ⋯)]
𝜋𝜔02 2
(13)
where 𝑉𝑑𝑒𝑡 (𝑥, 𝑡) is composed of AC and DC portions:
𝑉𝐷𝐶 = 𝑅𝐺𝑃0 − 𝜎𝑒𝑥𝑡
𝑎0
≅ 𝑅𝐺𝑃0
𝜋𝜔02
(14)
𝑉𝐴𝐶 (𝑥, 𝑡) = 𝜎𝑒𝑥𝑡
2𝑅𝐺𝑃0
(𝑏1 sin(2𝜋𝑓𝑡) + 𝑎2 cos(4𝜋𝑓𝑡) + ⋯).
𝜋𝜔02
In the SCSMS experiment, the AC signal is fed into a lockin amplifier that has an rms voltage output.
Since the 1f component [𝑏1 sin(2𝜋𝑓𝑡)] is sinusoidal, the rms amplitude is given by 𝑏1 ⁄√2. Therefore,
1𝑓
the Lock-in voltage [𝑉𝑙𝑜𝑐𝑘𝑖𝑛 (𝑥)] is proportional to the Fourier b1 coefficient, and equals
5
1𝑓
𝑉𝑙𝑜𝑐𝑘𝑖𝑛 (𝑥) = 𝜎𝑒𝑥𝑡
2𝑅𝐺𝑃0 𝑏1
√2𝜋𝜔02
= 𝜎𝑒𝑥𝑡
√2𝑉𝐷𝐶 𝑏1
.
𝜋𝜔02
(15)
The extinction cross section of a particle can be computed from Equation 15, resulting in
1𝑓
𝜎𝑒𝑥𝑡 =
𝑉𝑙𝑜𝑐𝑘𝑖𝑛 (𝑥𝑚𝑎𝑥 )𝜋𝜔02
√2𝑉𝐷𝐶 𝑏1
.
(16)
where 𝑥𝑚𝑎𝑥 is determined empirically by stepping the particle through the focus until a maximum lockin
signal is achieved (see Figure 3, main text).
II. TEM analysis of Au NP ensembles:
Five Au NP ensembles were purchased from Nanopartz and analyzed with a transmission
electron microscope. Representative TEMs and diameter sizing histograms are shown below. The
number of analyzed particles (N) is 55 for each ensemble.
6
7
Figure S3: TEM analyses of the five ensembles used in the optical study. The particles were purchased
from Nanopartz. Each panel shows the diameter distribution on the left along with a representative TEM
image on the right.
8
III. Comparison of literature Au NP extinction cross sections
Figure S4: Literature ext-values for different sized Au NPs compared to ext-values obtained in this
study. The dotted green line and black dashed line are Mie theory predictions for refractive indices of nm
= 1.5 and 1.0, respectively.
IV. Estimation of noise in extinction spectra
A spectrum’s noise is found by taking 1/3 the peak-to-peak amplitude of noise fluctuations in its red
portion (i.e. >650 nm). This is an estimation based on a normal distribution of noise, where 1/3 gives ~1
standard deviation (s) about the mean (𝑥̅ ).
9
Figure S5: Representation of a normal noise distribution about an average value (𝑥̅ ). 1/3 the peak-topeak value for observed noise fluctuations gives 1 standard deviation about the mean, which accounts for
68.2% of the total noise.
V. Low power extinction measurements
Figure S6: Extinction spectrum of an individual d~13 nm Au NP at low excitation intensity (200 Wcm-2).
This spectrum was acquired using two identical, balanced low-noise/high-gain photoreceivers for the
reference and signal channels (as described in the main text).
10
VI. Suspending/debundling SWCNTs:
Previously published methods were used to suspend/debundle the SWCNTs.1 In brief, 2 mg of (6,5)
rich CoMoCat SWCNT powder (Sigma Aldrich/SWeNT) was added to 10 ml of a 20 mg/mL solution of
sodium cholate hydrate (Sigma Aldrich). This solution was agitated with a horn sonicator (Sonifier 350)
for 45 minutes at 15% power output.
The temperature was controlled by submerging the sample
container in an ice bath. The resulting black suspension was then centrifuged at 18516xg for 1 hour,
whereupon the top 10% of the supernatant was drawn off and stored as the concentrated stock solution.
This was then diluted by ~10x with 20 mg/mL sodium cholate hydrate solution to achieve a dilute, clear
suspension. The solution was briefly sonicated (5 seconds) in a bath sonicator prior to drop casting onto
the microscope coverslip.
Figure S7: Ensemble linear absorption spectrum of a (diluted) stock SWCNT solution.
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VII. SWCNT literature cross section values
(6,5) Chirality
Transition
E11
E11
E22
E22
E22
E22
Cross section (cm2m-1)
8 x 10-13
3 x 10-13
4 x 10-13
9 x 10-14
5.5 – 11.5 x 10-13
3.7 x 10-12
(8,3) Chirality
This work
This work
E11
E22
4 x 10-13
5 x 10-13
(18,5) Chirality
D. Christofilos, et al., J. Phys. Chem. Lett. 3,1176 (2012)
E33
3.5 x 10-12
Paper
This work
L. J. Carlson, et al., Nano Lett. 7, 3698 (2007)
This work
L. J. Carlson, et al., Nano Lett. 7, 3698 (2007)
S. Berciaud, et al., Phys. Rev. Lett. 101, 077402 (2008)
L. Oudjedi, et al., J. Phys. Chem. Lett. 4, 1460 (2013)
1
Y. Tan, and D. E. Resasco, J. Phys. Chem. B 109, 14454 (2005).
12