Microscope Basics

Rays from an infinity objective are only parallel when from a single point;
with increasing distance between objective and field lens,
a larger field lens is needed to avoid vignetting:

An objective's back focal plane is where parallel rays entering that objective focus.

Lens formulae

Magnification: Classic compound microscope

Nikon's extensive Interactive Tutorials

Objectives: finite vs infinity, corrections, N.A., image circle

Zerene focus stacking

1:4 to 8:1 magnification FAQ

Compound microscope rays

Conjugate Planes

8 interleaved focal planes:: Specimen image at: (F1)field diaphragm, (F2)stage, (F3)intermediate image plane, (F4)retina (or camera sensor); Source lamp filament image (A1) at: (A2)condenser aperture, (A3)objective back focal plane, (A4)behind pupil

Worth watching: Abbe's experiments and conjugate planes

Finite vs infinity objectives; corrections, N.A., image circle

To prime your thinking about lenses and ray diagrams, review this explanation by Bill Otto.

The resolving power of an objective on the subject depends only on its N.A., not magnification
You can use a 20x 0.75 infinity objective at 40x with a 400mm tube lens,
with no degradation relative to a purpose-built 40x 0.75 with a 200mm tube lens.
At their specified focal length, objectives vary by useful field circle diameter.
Before 1980, 18mm was typical; modern Nikon Plan objectives can be 22mm or more.

Finite-conjugate microscope system with standardised tube length and infinite-conjugate microscope system with standardised tube lens

Finite-conjugate microscope system vs. infinite-conjugate microscope system with tube lens.
from: Systematic design of microscope objectives
The vertical dotted line where rays cross at right end of tube length is called the objective's rear conjugate.
To the extent that it is, a finite objective's correction is designed for that tube length,
keeping in mind than many objectives also depend on both slide cover glass and eyepiece for correction.

More explanation

The upper diagram is supposed to represent a traditional (RMS == Royal Microscope Society) microscope.
"tube length" is approximately the objective's focal length for its specified magnification,
if that objective was a simple convex lens.
The lower diagram, which is nominally about more modern microscopes,
is arguably also more accurate for some "finite" microscopes,
if Objective + Infinity space + Tube lens are considered together as a compensated "finite" objective.
While infinity objectives may be used with a tube lens of any focal length,
specified magnification depends on that focal length.

Contrast and non-image-forming light.

While lenses focus images on your retina or other sensor,
they do not prevent other photons from also stimulating sensors.
When viewing a three-dimensional scene, perhaps focusing on a near object,
photons scattered from more distant objects may also land on the same photosensors.
Properly (Köhler) aligned conjugate image and illumination planes also improve contrast.
Some non-image-forming light can be blocked by an iris diaphragm, as in this diagram:
light field microscope diagram

Ignore that Microlens Array. Relay part is afocal photography, where Field Lens is the eyepiece or ocular.

Lens formulae

simple

1/f = 1/do + 1/di {1} di becomes f for infinite do

m = di/do {2} zero magnification for lens focused @ infinity
magnification change by focus distance

f = (d2 - d1)/(m2 - m1); {3} alternatively:

d2 = d1 + f*(m2 - m1)

m2 = m1 + (d2 - d1)/f
magnification for classic (RMS) compound microscope

m = (L/fo)*(D/fe),
...where:

m = magnification

L = tube length (160mm)

D = normal vision relaxed distance (250mm)

f = focal length

fo = objective focal length

fe = eyepiece focal length

di = lens to image distance

do = lens to object distance
For 160mm tube length, a 10x objective has 16mm focal length
and a 10x eyepiece has 25mm focal length.
For infinity scopes, substitute "tube lens focal length" for "tube length".
Olympus infinity objectives expect 180mm tube lens focal length;
Nikon finite CF BD and M Plan objectives expect 210mm tube length.

Aberrations

Lens aberrations exist for 2 main reasons: incorrect geometries and changes in diffraction with wavelength.

Spherical lens surfaces are relatively easy to produce, but not ideal.
Combining different dispersions and refractive indices can reduce lens aberrations.

Compensate vs correct

correction == positive correction in a downstream optic for upstream optic deficiency.
compensation == inverse correction in a downstream optic to cancel upstream optic over correction.

Chromatic Aberrations (CA)

Correcting optics display blue interior CA and yellow exterior CA; reversed in compensating optics.
Correcting aberrations is typically accomplished by combining different simple lenses.
In traditional finite compound microscopes, objective aberrations were typically corrected in eyepieces,
with the amount of correction differing among manufactures, with mismatches problematic for N.A. >0.6 or so.
AO, Reichert, Zeiss and Leica infinity objective are corrected in tube lenses, typically located near heads' bottom port.
Nikon CF and modern Olympus infinity objectives are fully corrected, so useful with e.g. telephoto camera lenses.

Aperture

effective

Coupled lenses, stopped in the front, with the rear lens focused at infinity: m * lens aperture
Single lens, focused by extension: (m+1) * lens aperture
Teleconverter factor x inserted between camera and all other optics: x * lens aperture
Microscope objectives used as designed: m / (2 * N.A.)
equivalent e.g. "How does a 4X N.A. 0.1 objective compare to an f/whatever macro lens?"
f=1/(2*N.A.) is not a bad approximation.
A better approximation would be f=1/(2*N.A.) * M/(M+1), where M is rated magnification.
numeric: N.A. = n * sin(α), where n is (1.0 for air) index of refraction
"pupil ratio" compensates effective aperture for adding extension

aperture vs N.A. : N.A. = 1/(2 * f/#)
f/# 1.2 1.4 1.8 2 2.8 4 5.6 8 11 16
N.A. .4167 .357 .417 .25 .1786 .125 .0893 .0625 .045 .03125

Canon EF sensor-to-flange depth: 44mm

Add to extension tubes when testing lens focal lengths

Condensers: achro, aplanatic, Abbe; finite vs infinity

Tweaking Abbe condensers
resolution = 1.22 * wavelength / (Objective_NA + Condenser_NA)
Abbe flaws begin to bother above N.A. 0.6-7; elevate an Abbe to optimize filling the objective's back aperture.

Depth-of-Focus scaling

DoF2 = DoF1 * (f/#2/f/#1) * (m1/m2)**2

Diopter vs focal length: divide into 1000mm, e.g. `diopter = 2 for 500mm fl`

Zerene macro step size tables: magnification vs frame width, magnification vs aperture

magnification m = sensor width / frame width

DoF (mm) = 0.0022*(((m+1)*f/#)/m)**2

DoF (mm) = 0.00055/(N.A.**2)

Zerene landscape focus tables

1:4 to 8:1 magnification FAQ

Objective standards - DIN: 45mm parfocal distance

Finite tube length error impact on aberrations

N.A. 0.25 160 TL immersion objectives tolerate > 200mm tube length error.
N.A. 0.65 160 TL immersion objectives tolerate <= 5mm tube length error.

Tolerance goes inversely as N.A.^4:

gnuplot
More tube length discussion: Raynox DCR-150 tube assembly with flocking

Coverglass thickness error impact vs N.A.

I speculate that zero coverglass (metallurgical) objectives might be usefully more tolerant than coverglass-corrected.
Systematic impact suggests that modest aberrations (e.g. from wrong slide coverslip)
could be mitigated by deliberately changing tube length...
This would provoke magnification change and refocusing inconvenience.

Correcting eyepieces or tube lenses

Unlike e.g. Nikon or Olympus, Zeiss and Leica infinity objectives want proprietary tube lens corrections.

The right combination of objective and eyepiece

Higher objective magnifications are increasingly liable to optical aberrations,
but greatly reduced in modern larger and more complex infinity objectives,
while earlier systems applied finite objective corrections in compensating eyepieces.
Even highly regarded apochromatic finite objectives were undercorrected for lateral color aberrations.
Perhpas lower power finite objectives have aberrations deliberately introduced for compatibility...?

cmtalb01 tested correction eyepiece and 40x Zeiss objective combinations:

Notes:

"KPL" and "CPL" are Zeiss; "CPL" is "clinical plan" = good enough for eveyday routine use
"C5" is a Zeiss C5X eyepiece recommended for achromats, not more highly corrected objectives.
FK and NFK are Olympus
CFW eyepiece is Nikon (no corrections or compensations)
"DIC" is a Zeiss Epiplan objective;
relatively wide compatibility suggests that 0 coverslip objectives provoke fewer aberrations.
Hoff M is (probably Nikon CF-based) Hoffman modulation 40X
"Neo" is Zeiss Neofluar
"HD" is Zeiss Epiplan darkfield
"Phase" is probably Zeiss (since compatible only with Zeiss KPL eyepiece)
With phase contrast microscopy optimized for green light, chromatic aberrations are less of an issue.

As might be expected, a CFW eyepiece (applying no corrections) worked poorly with most Zeiss,
but OK with a (perhaps Nikon CF-based) Hoffman modulation 40X.

maintained by blekenbleu