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Phase imaging and detection in pseudo-heterodyne scattering scanning near-field optical microscopy measurements

Camilo Moreno, Javier Alda, Edward Kinzel, and Glenn Boreman

Open Access Open Access

Abstract

When considering the pseudo-heterodyne mode for detection of the modulus and phase of the near field from scattering scanning near-field optical microscopy (s-SNOM) measurements, processing only the modulus of the signal may produce an undesired constraint in the accessible values of the phase of the near field. A two-dimensional analysis of the signal provided by the data acquisition system makes it possible to obtain phase maps over the whole [0, 2π) range. This requires post-processing of the data to select the best coordinate system in which to represent the data along the direction of maximum variance. The analysis also provides a quantitative parameter describing how much of the total variance is included within the component selected for calculation of the modulus and phase of the near field. The dependence of the pseudo-heterodyne phase on the mean position of the reference mirror is analyzed, and the evolution of the global phase is extracted from the s-SNOM data. The results obtained from this technique compared well with the expected maps of the near-field phase obtained from simulations.

© 2017 Optical Society of America

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Corrections

3 February 2017: A correction was made to Refs. 2 and 5.

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Figures (8)
Fig. 1.
Fig. 1. Schematic of s-SNOM system with interferometer leg dithered for pseudo-heterodyne detection and a wire-grid polarizer for cross-polarized detection (BS, beam splitter; WGP, wire-grid polarizer; QWP, quarter-wave plate; LIA, lock-in amplifier; MCT, HgCdTe detector; AFM, atomic force microscope).

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Fig. 2.
Fig. 2. Maps of the topography and X and Y components of the near-field signal extracted from the first and second sidebands in pseudo-heterodyne detection mode for data set A. The angle of the signal, α , was set to zero before the measurement.

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Fig. 3.
Fig. 3. Data cloud of the signals registered for the (a, c, e) first and (b, d, f) second sidebands. The original data are shown in plots (a) and (b). The signals from the structure are plotted in (c) and (d). In these plots, the signals are self-centered, and the straight lines in (c) and (d) represent the directions of the maximum and minimum variance of the data. The maximum-variance direction is slightly misaligned with respect to the X axis. Figures (e) and (f) represent the original signal rotated to align data along the maximum-variance direction and centered to the mean of the data obtained from the substrate.

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Fig. 4.
Fig. 4. (a, c, e, g) Near-field modulus and (b, d, f, h) phase, corresponding to the square patches structures for data set A. Plots (a) and (b) correspond to the results of Eqs. (2) and (3) for the unsigned modulus of the signal. The results obtained from the method proposed in this paper are shown in plots (c) and (d), where the field is obtained from the X component of the signals. Plots (e) and (f) are obtained from component Y . Finally, these experimental results are compared with the simulations obtained from HFSS [see plots (g) and (h)] for the given structures under the same illumination that was used in the experiment.

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Fig. 5.
Fig. 5. Data cloud of the signals registered for the (a, c) first and (b, d) second sidebands for data set B. The original data are shown in plots (a) and (b). In this case, the maximum variance direction is largely misaligned with respect to the X axis. Figures (c) and (d) represent the original signal rotated to align the data along the maximum-variance direction and centered to the mean of the data obtained from the substrate.

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Fig. 6.
Fig. 6. (a) Near-field modulus and (b) phase, corresponding to the square-patches structures for data set B. Plot (c) corresponds to the results of Eq. (3) for the unsigned modulus of the signal.

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Fig. 7.
Fig. 7. Left: distribution of the near field for a specific position of the reference mirror. Each point defines the modulus and phase of the electric field at a given location on the sample. Right: angular representation of the normalized function F ( θ ) for the mirror position presented in the left plot.

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Fig. 8.
Fig. 8. Calculated phase, Ψ G , for different positions of the reference mirror. The origin of Ψ G is arbitrary, but it can be related to the position of a predetermined interference situation when the mirror is not vibrating.

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Tables (1)

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Table 1. Rotation Angle, β , and Relative Weight, w max , along the Maximum Variance Direction for the Two Sidebands (Subscripts 1 and 2)

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Equations (7)

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E R = ∑ m ρ m   exp ( i m f mirror t ) ,
E z ( x , y ) = κ [ S 2,2 ( x , y ) − i S 2,1 ( x , y ) ] exp [ i Ψ G ] ,
| E z ( x , y ) | = κ S 2,2 2 ( x , y ) + S 2,1 2 ( x , y ) ,
φ ( x , y ) = Ψ G − Φ ( x , y ) ,
Φ ( x , y ) = tan − 1 [ S 2,1 ( x , y ) S 2,2 ( x , y ) ] ,
w max = σ max 2 σ max 2 + σ min 2 ,
F ( θ ) = 1 M [ θ , θ + Δ θ ) ∑ j ∈ [ θ , θ + Δ θ ) | E z , j | ,
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