SME
Microbiology · Problem type
Resolution and magnification
Use this for “What is the smallest detail a 1.25 NA objective resolves in 550 nm light?”
Diagram
d
=
0.61λ
NA
=
0.61
×
550
1.25
=
268
nm
Total magnification = 100 × 10 = 1,000×
Two points are resolved when the gap is at least d: then the brightness dips between them.
This is an example. Type your own values to start a new problem.
d (nm)
=
0.61
×
λ (nm)
NA
g
Distance between two points
example · nm
k
Gap ÷ resolution limit
worked out (not typed)
Mobj
Objective
example · ×
Mₑᵧₑ
Eyepiece
example · ×
M
Total magnification
calculated · ×
gₛₑₑₙ
Gap as seen in the eyepiece
worked out (not typed) · μm
Formulas
d
=
0.61
×
λ
NA
268.4
=
0.61
×
550
1.25
k
=
g
d
1.1177
=
300
268.4
M
=
Mobj
×
Mₑᵧₑ
1,000 = 100 × 10
gₛₑₑₙ
=
g
×
M
1,000
300
=
300
×
1,000
1,000
Assumptions
•
Each point of light blurs to an Airy disk; d, the radius of its first dark ring, is the resolution limit.
•
Two points are just resolved when one’s peak sits on the other’s first dark ring (the Rayleigh criterion).
•
NA = n sin θ: immersion oil (n = 1.52) lets an objective pass NA above 1.
•
Magnifying past the limit makes the blur bigger, not the detail sharper.
Step-by-step
WE KNOW
λ = 550 nm, NA = 1.25, g = 300 nm, Mobj = 100×, Mₑᵧₑ = 10×
FIND
resolution limit (d), gap ÷ resolution limit (k), total magnification (M), gap as seen in the eyepiece (gₛₑₑₙ)
Step 1 · Find resolution limit (d)
Use d = 0.61 × λ ÷ NA
The Rayleigh limit: shorter light or a wider cone of it resolves finer detail.
d
=
0.61
×
550
1.25
d
=
335.5
1.25
d
=
268.4
nm
Step 2 · Find gap ÷ resolution limit (k)
Use k = g ÷ d
At 1 or more the two points are resolved.
k
=
300
268.4
k
=
1.1177
Step 3 · Find total magnification (M)
Use M = Mobj × Mₑᵧₑ
The eyepiece magnifies the objective’s image again.
M
=
100
×
10
M
=
1,000×
Step 4 · Find gap as seen in the eyepiece (gₛₑₑₙ)
Use gₛₑₑₙ = g × M ÷ 1,000
Magnifying makes the gap look bigger (1000 nm in a μm), but blurred disks grow with it.
gₛₑₑₙ
=
300
×
1,000
1,000
gₛₑₑₙ
=
300,000
1,000
gₛₑₑₙ = 300 μm
Check
Put the numbers back in. Both sides should match.
268.4
=
0.61
×
550
1.25
✓
1.1177
=
300
268.4
✓
1,000 = 100 × 10 ✓
300
=
300
×
1,000
1,000
✓
Topic check
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