Fig 1.
Schematic of idealized imaging system and focal shift.
(A,B) When the sample and the objective have difference refractive indices, moving the sample closer to the objective does not move the effective focal plane the same distance. Brown dashed line indicates the z’ axis and z’ = 0 at the surface of the objective. In this schematic, F and ΔZ have opposite signs, corresponding to a situation of n2 < n1. (C) Plots of intensity of electric field along the optical axis z’ for different positions of the n1 − n2 interface (Eq 1, NA = 1.2). Colors indicate sample positions from 150 μm (red) through 147 μm (green) to 145 μm (purple). (D) Focal shifts are proportional to sample displacements. Colored circles refer to peak positions from panel C. Darker grays indicate larger NAs (0.13, 1.0, 1.2, 1.3).
Fig 2.
Relative focal shift as measured as the distance between beads of different sizes.
(A) Example images with a sample of three types of beads (100 nm TetraSpeck, 510 nm Dragon Green, 1040 nm Dragon Green) are shown at different sample positions. (B) Brenner gradients for three example beads highlighted in panel A. (C) Box and whisker plots of the full-width at half-maximum Brenner gradient (brown) and peak intensity (green). (D) Histograms of relative distances between beads and the plane of the smallest beads. (E) Sample motion needed to refocus from one size bead to another is proportional to the difference in their sizes. α is the slope of this line. Colors as in panel D. x error bars are standard deviations of observed positions. y error bars are 5% relative deviation in bead diameter. Solid line is the fit y = αx and the dashed lines are the 95% C.I. for α. (F) Standard deviation of bead positions, measured by Brenner gradient (brown) and peak intensity (green). 99% C.I. for σ is calculated from a bootstrap analysis and displayed as the error bars.
Fig 3.
Fluorescent images showing the disparity between slices in a single focal plane and along the focal dimension.
Scale bars are 1 μm. (A-E) E. coli cell stained with a membrane dye (FM 4–64). (F-J) 1 μm sphere with a fluorescent ring stain. (A,F) xy slice (B,G) xz slice showing the apparent elongation of the object along the focal axis. (C,H) Active contour fit to the ridge of maximal intensity in red. (D,I) Stretched circle fit in blue r2 = (αz)2 + x2. (E,J) xz slice shown after scaling z by α.
Fig 4.
Polynomial extrapolation of relative focal shift as a function of NA and n2.
(A) αpoly as a function of NA and n2. The colormap (inset) gives the value of α. The TIR region is marked by a dashed gold line. Colored arrowheads indicate where the 1D slices through this surface are taken for panels C and D. (B) The order of the polynomial extrapolation is chosen as the lowest order whose maximal error (grape) is less than the maximal difference between α646 and α489 (cocoa). (C-D) α evaluated by the theory in [2] at various values of NA and n2 (symbols) along with the global fit to Eq 3 (lines). Symbols that corresponding to TIR systems are shown as faded circles. (C) α as a function of refractive index for three different values of numerical aperture (1.00 in turquoise, 1.38 in olive, 1.48 in orchid). Also included is the n2/n1 approximation (gray—⋅⋅). (D) α as a function of numerical aperture for three different values of n2 (1.33 in pistachio, 1.38 in goldenrod, 1.43 in periwinkle). (E) Experimentally observed α for imaging system of varying refractive index. Refractive index of the medium was increased by including glycerol (cornflower) or sucrose (dark red). Two copies of the same model of objective were measured, results from one are filled symbols and the other are open. Evaluating the αpoly with an NA of 1.49 is shown in charcoal. α error bars are 90% CI. n2 error bars reflect changes in additive concentration by ± 1.5%. (F) Experimentally observed αmultiBead as a function of NA for imaging systems with various objectives using the multiple bead sizes method.
Fig 5.
Constant α near the the coverslip.
(A) Apparent α for various sample depths as predicted by Eqs 1–2. (B) αtwoPoint plotted against the average radius of the two bead sizes that went into the measurement.
Table 1.
Properties of objectives used to test the versatility of the multiple bead method.
Table 2.
Summary of a variety of methods used to measure relative focal shift.
Values are reported for our highest NA objectives, NA 1.49, imaging into an aqueous environment, n2 = 1.33.