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^{1}, Cheol-Joo Kim

^{2}, Zenghui Wang

^{2}, Moon-Ho Jo

^{3}and Jiwoong Park

^{2,4,a)}

### Abstract

We measure absolute optical absorption cross-sections of one- (1D) and two-dimensional (2D) nanostructures using a focused laser beam while varying the numerical aperture (NA) of the focusing lens. We find the optical absorption deviates at higher NA. In the high NA regime, absorption by graphene decreases from 2.2% to below 1.8%; for Ge nanowires, it decreases from an expected value by a factor of 1.2. We explain this using the depolarization effect at the focal spot and conclude that these corrections allow for accurate quantitative measurements of optical and optoelectronic processes in 1D or 2D nanostructures.

We thank M. Wojcik for providing the graphene samples. This work was supported by NSF CAREER Grant (DMR-0748530). M.H.J. acknowledges the support the Nano Original & Fundamental Technology R&D Program (2010-0019195) and the Mid-career Researcher Program (2010-0027627).

### Key Topics

- Polarization
- 20.0
- Nanostructures
- 17.0
- Optical absorption
- 17.0
- Graphene
- 12.0
- Nonlinear acoustics
- 11.0

## Figures

Measurement scheme and calculated depolarization effect at the focal plane. (a) Linearly polarized light (785 nm) is introduced into an oil-immersion objective lens with an adjustable numerical aperture (upper blue disk), with the polarization direction shown as a blue arrow. The light transmitted through the substrate which has various nanostructures resting on the surface is collected by an oil-immersion lens (lower blue disk) with a fixed numerical aperture (NA = 1.4). The initial polarization before the lens is along the *x*-direction, and the *z*-direction is defined as being normal to the sample. Two representative semiaperture angles are shown here. Focusing the light with the larger semiaperture angle tilts the polarization, introducing a measurable *z*-component. (b) Calculated intensities of the *x* and *z* components of the electric field incident upon the sample focal plane are shown fordifferent SAs. All intensities at different SAs are normalized to the *x*-component.

Measurement scheme and calculated depolarization effect at the focal plane. (a) Linearly polarized light (785 nm) is introduced into an oil-immersion objective lens with an adjustable numerical aperture (upper blue disk), with the polarization direction shown as a blue arrow. The light transmitted through the substrate which has various nanostructures resting on the surface is collected by an oil-immersion lens (lower blue disk) with a fixed numerical aperture (NA = 1.4). The initial polarization before the lens is along the *x*-direction, and the *z*-direction is defined as being normal to the sample. Two representative semiaperture angles are shown here. Focusing the light with the larger semiaperture angle tilts the polarization, introducing a measurable *z*-component. (b) Calculated intensities of the *x* and *z* components of the electric field incident upon the sample focal plane are shown fordifferent SAs. All intensities at different SAs are normalized to the *x*-component.

The experimentally measured absorption in graphene depends on semiaperture illumination angle. (a) Absorption of light at the same area of the sample for two different semiaperture angles. The absorbance of the substrate was set to zero following uniform background correction. Scale bar, 5 m. (b) Experimentally measured (black squares) and theoretical curve (red line) of absorption of graphene as a function of semiaperture angle. The theoretical curve takes into account the depolarization effect at the focal plane obtaining the fitted tangential absorption of light to be for graphene. Inset: Laser beam diameter is smaller than the area of the graphene sheet, which is assumed in the calculation.

The experimentally measured absorption in graphene depends on semiaperture illumination angle. (a) Absorption of light at the same area of the sample for two different semiaperture angles. The absorbance of the substrate was set to zero following uniform background correction. Scale bar, 5 m. (b) Experimentally measured (black squares) and theoretical curve (red line) of absorption of graphene as a function of semiaperture angle. The theoretical curve takes into account the depolarization effect at the focal plane obtaining the fitted tangential absorption of light to be for graphene. Inset: Laser beam diameter is smaller than the area of the graphene sheet, which is assumed in the calculation.

Dependence of measured absorption versus illumination semiaperture angle in Ge nanowires. (a) Upper: AFM image of a representative Genanowire (d 65 nm). Scale bar, 1 m. Lower: Measured absorption of light of the same nanowire for two different semiaperture angles. Scale bar, 0.5 m. (b) Experimentally measured (black squares) and calculated (blue and red curves) absorption for various semiaperture angles of illumination. The blue curve considers only the Abbe diffraction effect in the absence of depolarization. The red curve, however, includes the depolarization effects. Both calculations for this nanowire use in the portion of the nanowire being illuminated. The latter uses . Inset: Schematic illustrating that the nanowire diameter is smaller than the beam diameter. Smaller semiaperture angles lead to larger laser spot diameters (left) and vice versa (right), with the *z* component (shaded area) only being significant in the latter.

Dependence of measured absorption versus illumination semiaperture angle in Ge nanowires. (a) Upper: AFM image of a representative Genanowire (d 65 nm). Scale bar, 1 m. Lower: Measured absorption of light of the same nanowire for two different semiaperture angles. Scale bar, 0.5 m. (b) Experimentally measured (black squares) and calculated (blue and red curves) absorption for various semiaperture angles of illumination. The blue curve considers only the Abbe diffraction effect in the absence of depolarization. The red curve, however, includes the depolarization effects. Both calculations for this nanowire use in the portion of the nanowire being illuminated. The latter uses . Inset: Schematic illustrating that the nanowire diameter is smaller than the beam diameter. Smaller semiaperture angles lead to larger laser spot diameters (left) and vice versa (right), with the *z* component (shaded area) only being significant in the latter.

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