Measurement of complex optical susceptibility for individual carbon nanotubes by elliptically polarized light excitation

The complex optical susceptibility is the most fundamental parameter characterizing light-matter interactions and determining optical applications in any material. In one-dimensional (1D) materials, all conventional techniques to measure the complex susceptibility become invalid. Here we report a methodology to measure the complex optical susceptibility of individual 1D materials by an elliptical-polarization-based optical homodyne detection. This method is based on the accurate manipulation of interference between incident left- (right-) handed elliptically polarized light and the scattering light, which results in the opposite (same) contribution of the real and imaginary susceptibility in two sets of spectra. We successfully demonstrate its application in determining complex susceptibility of individual chirality-defined carbon nanotubes in a broad optical spectral range (1.6–2.7 eV) and under different environments (suspended and in device). This full characterization of the complex optical responses should accelerate applications of various 1D nanomaterials in future photonic, optoelectronic, photovoltaic, and bio-imaging devices.

Where the first −i π 2 term is caused by Gouy's phase shift, and the second −i π 2 term is from the phase difference between the fast and slow axis of a quarter-wave plate. The nanotube scattering electric field is polarized along the nanotube direction due to its strong depolarization effect, the scattering field ( NT ) amplitude would be NT L =̃L =̃i n [e −i π 2 cos cos ( π 4 + ) + sin cos ( Here represents an efficiency coefficient (details in Supplementary Note 2). Then, we could derive the nanotube scattering field after P2 ( s L ) as The approximation at last is reasonable because is set as a small angle ( ≤ 4°).
In the second case, to generate a right-handed elliptically polarized light, we keep the angle between the wave plate and the second polarizer (P2) at θ. In this case, the local field would ].
Where R represents the transmitting light field after P2, s R represents the nanotube scattering field after P2 in this case. Then we could derive 10 / 13

Supplementary Note 2. Determine the efficiency constant
The supercontinuum with a Gaussian spatial profile can be described as Where and are the coordinates of the center position of the focus and R is a measure of the beam size. So energy density function of laser can be described as For a 1D nanotube (with a small diameter d) along direction and positioned at , the ratio between the total scattering intensity and incident light intensity over the nanotube length is 2-5 Here is the angular frequency of light, is the speed of light and d is the diameter of nanotube.
Where represents the reflection coefficient calculated from Fresnel equations and 1 + represents the local field experienced by the nanotube.
Then we could derive Thus, we could see that the optical signal will be universally increased by (1 + ) 2 / from transmission to reflection configuration.

Supplementary Note 4. Estimate the sensitivity of our technique
This technique is based on the manipulation of interference between incident left-(right-) handed elliptically polarized light and materials' scattering light, physically this technique is not limited to CNTs, but establishes a general analytical means for the entire class of 1D materials.
The only limitation should come from the signal-to-noise level of other 1D systems. Here we consider the detection limit of our technique and estimate how far our technique can go for other 1D materials with possible smaller signal. Firstly, according to the performance of our detector (linear CCD, Imaging Solution Group, LW ELIS-1024a-1394, 14 bit), the minimum contrast signal that can be measured is about 10 -4 (the average of sufficient data to minimize the random noise is needed). Secondly, considering ~20-100 times enhancement of the technique based on polarization manipulation, the detection limit of our technique could be ~10 -6 .