The Optics Laboratory
Group of Hans Hallen, Physics Department, North Carolina State University

Spatial Resolution

In NSOM, the spatial resolution is usually given by the aperture size of the probe. However, we have seen that carrier diffusion is important in these recombination lifetime studies. This might limit the maximum resolution. We turn to the data to see:

 

The image on the right (the part of the larger image outlined with the white square) is λ-square (1.15 micron square). It clearly shows subwavelength resolution. In fact, the resolution seems to be the aperture size.

• Question:

- Why do we obtain high resolution when carrier diffusion is so large?

- With D = 10 cm2/s and τ > 1 µs, diffusion lengths √(Dτ) > 30 µm.

Naively, this would suggest an optimal spatial resolution of tens of microns. We see why not with a simple model.

- Consider a simple case: τ=0 inside a sphere of radius R (carriers recombine immediately), τ=∞ outside (carriers never recombine). Symmetry allows us to cleave all of space in half to obtain the solution for a small, hemispherical defect at the surface -- something we might expect to be a reasonable model. The problem is exactly solvable. The profile n/n0 ~ 1 - R/r depends on R not D.

- The size of the defect determines the ultimate resolution, not diffusion. Point defects are small enough that the microscope resolution limit applies.

- D (diffusion) is reflected in the contrast.

- Diffusion reduces the contrast around the defect. In particular, the depression is shallower, so that better signal to noise is required to observe it. Realistic time constants further this smoothing.

 

• We now summarize in a model:

- The carriers, once formed locally by the visible light, rapidly diffuse. (diffusion calculations are in SPIE95.pdf and MRS96.pdf in papers)

- The diffusion profile reflects the local time constant.

- Resolution is provided by the NSOM through the infrared light, which images the diffusion profile. (see the APL)

- Estimate the contrast Δn/n ~ (Δr)2 Δ(τ-1) /D. (see MRS96.pdf)

- The measurements show that the contrast is adequate.

 

The ultimate resolution above was estimated as a spot size for an isolated point defect. What about more complicates structures? Are the internal features retained?

• Simulate Images, the carrier relaxation rate is 10-20

times faster within the F-letter (20X in top part).

• Background lifetime: 1 µs, frame size: 10 µm, Do = 1 cm2/s

• Scale the data range onto the color scale by 'scale.'

 

 D = 0, scale = 1

D = 0.02 Do, scale=1.05

D = 0.1 Do, scale=1.3

D = 0.25 Do, scale=1.7

D = Do, scale=3.2

Since the internal features are approximately the same size as the line width, they get washed out as diffusion is 'turned on.' Notice also how the contrast decreases with increased diffusion, making quantitative the earlier statements.

  •  More info is in the papers.

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