Biomedical Engineering Reference
In-Depth Information
Since.the.raw.phase.function.itself.is.hard.to.interpret,.it.is.useful.to.represent.the.phase.func-
tion. of. the. aberrated. wavefront. by. its. Zernike. mode. content.. The. Zernike. polynomials. are. a. set.
of.orthogonal.functions.over.the.unit.circle;.some.of.the.lower-order.modes.correspond.directly.
to. the. classic. aberration. terms,. as. for. instance,. astigmatism. and. coma.. It. is. also. convenient. to.
base. correction. schemes. of. adaptive. optics. systems. on. Zernike. modes. since,. in. many. cases,. it. is.
sufficient. to. correct. for. a. few. lower-order. Zernike. modes. only.. Because. of. the. orthogonality. of.
the.polynomials,.the.decomposition.of.the.phase.function.into.Zernike.mode.amplitudes.can.be.
written.as
1
1
2
π
∫
∫
.
M
=
ψ θ
( , )
r
Z r
( , ) d d
.
θ
r
θ
r
(6.7)
i
i
π
0
0
where.
Z
i
(
r
,θ).is.the.Zernike.polynomial.of.order.
i
,.and.
M
i
,.the.corresponding.amplitude.of.that.mode..
For.a.listing.of.lower-order.modes,.see.Table.4.1.of.Chapter.4.
Figure.6.3.shows.the.simulated.Zernike.mode.variations.of.the.aberrated.wavefront.while.“scan-
ning”.across.the.virtual.spherical.sample..he.simulations.(
Figure
.6.3a
.through.
6.3c
).difer.in.focus-
ing. depth.. he. plane. probed. in. Figure. 6.3a. contains. the. center. of. the. sphere. and. shows. the. lowest.
magnitude. of. aberrations,. and. no. tip. or. tilt. is. present. because. of. the. symmetry. conditions. for. the.
cone. of. rays. focused. to. this. plane.. For. an. intermediate. focal. position. between. center. and. bottom.
(
Figure 6.3b
)
.and.for.the.bottom.focal.plane.(
Figure.6.3c
)
,.higher-aberration.magnitudes.are.found;.
nonzero.values.for.the.tip/tilt.modes.and.slightly.changed.shapes.of.some.of.the.Zernike.aberration.
patterns.occur.
Mode 2 - tip
Mode 3 - tilt
Mode 4 - defocus
(a)
0.5
3
2
1
0
0
̶
1
̶
2
̶
3
̶
0.5
Mode 5 - astigmatism
Mode 6 - astigmatism
Mode 7 - coma
Mode 8 - coma
Mode 9
Mode 10
Mode 11 - spherical (1st)
Mode 12
Mode 13
Mode 14
FIGuRE 6.3 (See color insert.)
.Simulation.of.specimen-induced.aberrations.for.a.sphere.of.80.μm.in.diameter.
and.uniform.refractive.index.
n
1
.=.1.35,.embedded.in.water.(
n
0
.=.1.33).and.dry.lens.with.the.numerical..aperture.
(NA) =.0.5..he.virtual.sample.was.scanned.on.a.32.×.32.grid.covering.a.ield.of.view.of.100.μm.×.100.μm..For. each.of.
the.grid.positions,.a.wavefront.is.calculated.using.the.described.ray-tracing.method..hree.focal.depths.are shown:.
(a).center.of.the.cell—40.μm.above.bottom,.(b).intermediate.plane—20.μm.above.the.bottom,.and.(c)..bottom.of.the.
cell..Note.the.diferent.scales.among.the.simulations.
