Optical Microscope

1. Theory

• Definition: An optical microscope forms a magnified image of a sample with visible or ultraviolet light through an objective and tube lens, viewed by eye or camera (TU Dresden; Basler).
• Resolution: Diffraction limits resolution to about half a wavelength divided by the numerical aperture (Abbe) or 0.61 wavelength divided by NA (Rayleigh) (TU Dresden; Basler; KEYENCE).
• Numerical aperture: $NA=n\sin \alpha$ sets resolution, brightness and depth of field together (TU Dresden).
• Contrast: Bright field, dark field and differential interference contrast reveal different features (Dobbie).

2. Working Principle

• Objective and tube lens: Infinity-corrected objectives with magnification and NA engraved, such as 40x/0.95 (TU Dresden).
• Koehler illumination: Field and aperture diaphragms control illuminated area and illumination NA, separating imaging and illumination planes (Dobbie).
• Camera: The pixel size divided by magnification must be at most about one 2.3th of the resolution for Nyquist sampling (Basler).
• Immersion: Oil with index 1.515 raises NA to about 1.4 (TU Dresden).

3. Applications

• Wafer and die inspection: Particles, scratches and pattern defects, often in dark field (Dobbie for dark field; use my reading).
• Linewidth estimates: Coarse critical dimensions above the resolution limit (Basler).
• Topography: DIC for steps and polishing residues (Dobbie for DIC; use my reading).

Aperture and Resolution

Name	Formula	Related Formulae	Variables/Constants
Numerical aperture	$NA=n\sin \alpha$	$N{A}_{max}\approx 0.95(\text{air}),NA\approx 1.4(\text{oil},n=1.515)$	$NA:numericalaperture$ $\begin{array}{r}n:refractiveindexofthemediumbetween\\ lensandsample\end{array}$ $\begin{array}{r}\alpha :half-angleofthelightconeaccepted\\ bytheobjective;TUDresden:this\\ definition,oilindex1.515,examples\\ 1.515sin67degrees=1.4and1xsin72\\ degrees=0.95,objectivepairs12.5/0.3,\\ 25/0.3and0.65,40/0.65and0.95,90/1.32\\ oil;KEYENCE:samedefinition,n(air)=1\end{array}$ $\begin{array}{r}(computed:1.515sin67degrees=1.395\\ andsin72degrees=0.951)\end{array}$
Abbe and Rayleigh resolution	$d_{Abbe}=\frac{\lambda }{2NA},d_{Rayleigh}=\frac{0.61\lambda }{NA}$	$d_{Abbe}=\frac{\lambda }{N{A}_{obj}+N{A}_{cond}}(\text{transmitted light})$	$d:smallestresolvableseparation$ $\lambda :wavelength$ $\begin{array}{r}N{A}_{obj},N{A}_{cond}:objectiveandcondenser\\ apertures;TUDresden:\lambda /(2NA),0.5\lambda /NA\\ and0.61\lambda /NA;Basler:Abbelimitabout179\\ \mathrm{nm}for500\mathrm{nm}lightatNA1.4andabout\\ 200\mathrm{nm}inpractice,Rayleigh0.61\lambda /NAwith\\ atypicaldatasheetvalueof0.85\mu \mathrm{m}at\\ 550\mathrm{nm};KEYENCE:0.61\lambda /NAastheAiry-disk\\ radius;condenserform(myreading)\end{array}$ $\begin{array}{r}(computedat550\mathrm{nm}:NA0.95gives289\mathrm{nm}\\ (Abbe)and353\mathrm{nm}(Rayleigh);NA1.4gives\\ 196and240\mathrm{nm};the0.85\mu \mathrm{m}datasheetvalue\\ impliesNA0.39)\end{array}$
Ultraviolet inspection	$d\propto \lambda$	$d_{Rayleigh}\left(248\mathrm{nm},NA0.9\right)$	$\begin{array}{r}scalingofresolutionwithwavelength\\ (computedfromtheRayleighform);useof\\ deep-ultravioletobjectivesforfiner\\ features(myreading)\end{array}$ $\begin{array}{r}(computed:0.61x248\mathrm{nm}/0.9=168\mathrm{nm},\\ against373\mathrm{nm}for550\mathrm{nm}atthesameNA)\end{array}$

Depth, Sampling and Brightness

Name	Formula	Related Formulae	Variables/Constants
Depth of field	$d_{tot}=\frac{\lambda n}{N{A}^2}+\frac{n}{M\cdot NA}e$	$d_{tot}\propto \frac{1}{N{A}^2}(\text{wave-optics term})$	$d_{tot}:depthoffield$ $\begin{array}{r}M:totallateralmagnificationtothe\\ detector\end{array}$ $\begin{array}{r}e:smallestresolvabledetectordistance\\ (pixelsize);Microspectra:depthoffocus\\ n\lambda /N{A}^{2}asthewaveterm;thedetectorterm\\ (myreading);TUDresden:depthoffield\\ approximatelyinverselyproportionaltoN{A}^2\end{array}$ $\begin{array}{r}(assumed:\lambda =550\mathrm{nm},e=6.5\mu \mathrm{m};computed:\\ 100x/0.95gives0.68\mu \mathrm{m}and10x/0.30gives\\ 8.3\mu \mathrm{m})\end{array}$
Camera sampling	$\frac{p_{cam}}{M}\le \frac{d}{2K}$	$K\approx 1.1\text{to}1.3,2K\approx 2.3$	$p_{cam}:camerapixelsize$ $M:magnificationtothecamera$ $d:opticalresolution$ $\begin{array}{r}K:safetyfactor;Basler:Nyquistsampling\\ pitchofatmosthalftheresolution,K\\ typically1.1to1.3with1.15givingthe\\ industrymultiplierof2.3,workedexample\\ for2\mu \mathrm{m}defectsat550\mathrm{nm}needingNAat\\ least0.17(0.3chosen),0.87\mu \mathrm{m}effective\\ pixelsandabout6x,4xor2.5xfor5,\\ 3.45and2\mu \mathrm{m}pixels,andatableof\\ requiredpixelsizes(forexample2.8\mu \mathrm{m}at\\ 2x/0.10and4.6\mu \mathrm{m}at10x/0.30)\end{array}$ $\begin{array}{r}(computed:10x/0.30withRayleigh1.12\mu \mathrm{m}\\ allows4.9\mu \mathrm{m}pixels;withAbbe0.92\mu \mathrm{m},4.0\\ \mu \mathrm{m})\end{array}$
Brightness and useful magnification	$B\propto {\left(\frac{NA}{M}\right)}^2(\text{transmitted bright field})$	$M_{useful}\approx 500NA\text{to}1000NA$	$B:imagebrightness$ $\begin{array}{r}{M}_{useful}:rangeoftotalmagnificationabove\\ whichdetailisnotimproved(empty\\ magnification);TUDresden:light\\ transmissionscalingwithN{A}^2;\\ Microspectra:usefulmagnification500to\\ 1000timesNA;brightnessrule(my\\ reading)\end{array}$ $\begin{array}{r}(computed:100x/0.95isabout10times\\ dimmerthan10x/0.30;NA0.95supportsup\\ toabout950xtotalmagnification)\end{array}$

Contrast Methods

Name	Formula	Related Formulae	Variables/Constants
Bright field and dark field	$N{A}_{ill}<N{A}_{obj}(\text{bright field}),N{A}_{ill}>N{A}_{obj}(\text{dark field})$	$I_{DF}\propto \text{scattered light only}$	$\begin{array}{r}N{A}_{ill}:illuminationaperture;indarkfield\\ thedirectbeammissestheobjectiveso\\ onlylightscatteredbyparticles,\\ scratchesandedgesiscollected(Dobbie:\\ darkfieldcondenserNAlargerthanthe\\ objectiveNA)\end{array}$ $\begin{array}{r}(computed:a0.9NAdark-fieldringaround\\ a0.8NAobjectiveexcludesallspecular\\ reflectionfromaflatwafer)\end{array}$
Differential interference contrast	$\Delta \varphi =\frac{4\pi }{\lambda }\Delta h(\text{reflection})$	$I\propto {\sin }^2\left(\frac{\Delta \varphi +{\varphi }_{bias}}{2}\right)$	$\begin{array}{r}\Delta \varphi :phasedifferencebetweentwosheared\\ beams\end{array}$ $\Delta h:heightdifferenceacrosstheshear$ $\begin{array}{r}{\varphi }_{bias}:prismbias;Dobbie:Wollastonprism\\ splittingintotwoorthogonallypolarized\\ beams,phasedifferenceconvertedto\\ intensity;reflected-lightrelations\\ (computed)\end{array}$ $\begin{array}{r}(computed:a10\mathrm{nm}stepgives\Delta \varphi =0.23\\ radat550\mathrm{nm},enoughtosee)\end{array}$
Koehler illumination	$\text{field diaphragm}\leftrightarrow \text{sample plane},\text{aperture diaphragm}\leftrightarrow \text{back focal plane}$	$N{A}_{ill}\approx 0.7\text{to}0.8N{A}_{obj}(\text{typical setting})$	$\begin{array}{r}conjugateplanesforevenilluminationand\\ adjustableilluminationaperture;Dobbie:\\ Koehlerilluminationseparatingimaging\\ andilluminationplanes;typicalaperture\\ setting(myreading)\end{array}$ $\begin{array}{r}(computed:fora0.95objective,0.67to\\ 0.76illuminationNA)\end{array}$

Important Questions

Ques: What limits an optical microscope’s resolution?

Diffraction, through the wavelength and numerical aperture. A point images as an Airy disk of radius $0.61\lambda /NA$ (KEYENCE), and two points are just resolved when one lies on the other’s first minimum, the Rayleigh criterion; Abbe’s grating analysis gives $\lambda /(2NA)$ (TU Dresden; Basler). Basler gives about 179 $\mathrm{nm}$ for 500 $\mathrm{nm}$ light at NA 1.4, about 200 $\mathrm{nm}$ in practice. At 550 $\mathrm{nm}$, a 0.95 dry objective resolves 289 $\mathrm{nm}$ by Abbe and 353 $\mathrm{nm}$ by Rayleigh (computed). Magnification does not improve this: beyond about 500 to 1000 times NA, enlargement adds no detail (Microspectra). Shorter wavelengths help, which is why deep-ultraviolet inspection at 248 $\mathrm{nm}$ reaches 168 $\mathrm{nm}$ at NA 0.9 (computed). Reference: KEYENCE, ‘Numerical Aperture (N.A.)’ microscope glossary entry (fetch-tool summary); TU Dresden BioDIP, ‘The importance of Aperture in the Microscope’, teaching slides 06 Numerical Aperture (author not stated) (fetch-tool summary); Basler, ‘Microscopy resolution limits’ learning article (fetch-tool summary); Microspectra, ‘Microscope Formulas’ sheet: resolving power 0.61 lambda/NA, depth of focus n lambda/NA^2, depth of field lambda sqrt(n^2 - NA^2)/NA^2, useful magnification 500 to 1000 times NA (fetch-tool summary).

Ques: Basler’s table gives 0.92 microns for a 10x/0.30 objective, but its example gives 1.1 microns. Which is right?

Both, because they use different criteria. Basler’s required-pixel table lists 0.92 $\mu \mathrm{m}$ for 10x/0.30, which is the Abbe value $\lambda /(2NA)$ at 550 $\mathrm{nm}$, 0.917 $\mu \mathrm{m}$, while its worked example gives about 1.1 $\mu \mathrm{m}$ for NA 0.3, which is the Rayleigh value $0.61\lambda /NA$, 1.118 $\mu \mathrm{m}$ (computed). Its table entry for 2x/0.10, 2.8 $\mu \mathrm{m}$, also matches Abbe (2.75 $\mu \mathrm{m}$) rather than Rayleigh (3.36 $\mu \mathrm{m}$) (computed). The Rayleigh value is the more conservative and more common specification for incoherent imaging, so it is the safer choice for defect detection (my reading). The choice matters for camera matching: 10x/0.30 allows 4.9 $\mu \mathrm{m}$ pixels with Rayleigh but 4.0 $\mu \mathrm{m}$ with Abbe (computed). Reference: Basler, ‘Microscopy resolution limits’ learning article (fetch-tool summary).

Ques: Why does immersion oil raise the numerical aperture?

Because NA includes the refractive index of the medium. $NA=n\sin \alpha$, and in air $n=1$, so even a 72 degree half-angle gives only 0.95 (TU Dresden; computed 0.951). Oil of index 1.515 fills the gap between lens and coverslip, so the same geometry allows NA about 1.4: TU Dresden’s example is 1.515 times sin 67 degrees, 1.395 (computed). Oil also stops light from being refracted away at the glass-air surface, so more is collected. Light transmission scales with $N{A}^2$ (TU Dresden), so oil objectives are brighter at a given magnification. Wafer inspection rarely uses oil because it contaminates surfaces; dry objectives up to about 0.95 and ultraviolet wavelengths are used instead (my reading). Reference: TU Dresden BioDIP, ‘The importance of Aperture in the Microscope’, teaching slides 06 Numerical Aperture (author not stated) (fetch-tool summary); KEYENCE, ‘Numerical Aperture (N.A.)’ microscope glossary entry (fetch-tool summary).

Ques: Why is depth of field so small at high NA?

Because the wave-optics term falls as $1/N{A}^2$. Depth of field is approximately $\lambda n/N{A}^2+ne/(M\cdot NA)$, the second term from detector pixel size (Microspectra gives the wave term $n\lambda /N{A}^2$ as depth of focus; the detector term is my reading); TU Dresden notes the inverse-square dependence on NA. With 550 $\mathrm{nm}$ light and 6.5 $\mu \mathrm{m}$ pixels, a 100x/0.95 objective has only 0.68 $\mu \mathrm{m}$, while 10x/0.30 has 8.3 $\mu \mathrm{m}$ (computed). On a wafer with topography of a micrometre or more, high-NA images show only one level sharply, so focus stacking, confocal microscopy or a low-NA overview is needed. The trade-off is unavoidable: the NA that gives resolution also removes depth. Autofocus systems on inspection tools must therefore hold focus to a fraction of a micrometre. Reference: TU Dresden BioDIP, ‘The importance of Aperture in the Microscope’, teaching slides 06 Numerical Aperture (author not stated) (fetch-tool summary); Microspectra, ‘Microscope Formulas’ sheet: resolving power 0.61 lambda/NA, depth of focus n lambda/NA^2, depth of field lambda sqrt(n^2 - NA^2)/NA^2, useful magnification 500 to 1000 times NA (fetch-tool summary).

Ques: How should a camera be matched to the objective?

So that each resolution element spans at least about 2.3 pixels. Basler applies Nyquist sampling, effective pixel at most half the resolution, with a safety factor K of 1.1 to 1.3, giving the multiplier 2.3. Its example for 2 $\mu \mathrm{m}$ defects at 550 $\mathrm{nm}$ needs NA at least 0.17, chooses 0.3, and needs effective pixels of about 0.87 $\mu \mathrm{m}$, so about 6x for 5 $\mu \mathrm{m}$ pixels, 4x for 3.45 $\mu \mathrm{m}$ and 2.5x for 2 $\mu \mathrm{m}$. Too coarse a pixel loses resolution the optics provide; too fine a pixel wastes field of view and light. With 6.5 $\mu \mathrm{m}$ pixels and 100x, the effective pixel is 65 $\mathrm{nm}$, finer than needed for the 353 $\mathrm{nm}$ Rayleigh limit at NA 0.95 (computed). Reference: Basler, ‘Microscopy resolution limits’ learning article (fetch-tool summary).

Ques: When are dark field and DIC used on wafers?

Dark field for particles and scratches, DIC for shallow steps. In dark field the illumination arrives at angles outside the objective’s acceptance, so a flat wafer looks black and only light scattered by particles, scratches and pattern edges reaches the camera; a 0.9 NA ring around a 0.8 NA objective excludes all specular light (Dobbie: the darkfield condenser NA must exceed the objective NA; computed). Particles smaller than the resolution limit become visible as bright points, though not sized. Differential interference contrast splits the beam into two sheared beams whose phase difference, $4\pi \Delta h/\lambda$ in reflection, turns height differences into intensity; a 10 $\mathrm{nm}$ step gives 0.23 $\mathrm{rad}$ at 550 $\mathrm{nm}$ (Dobbie for the beam splitting; reflection form computed), visible after CMP or etch. Bright field remains best for colour and film-thickness contrast. Reference: I. Dobbie, ‘Brightfield techniques, contrast enhancement’, Micron lecture slides, University of Oxford: Koehler illumination separating imaging and illumination planes, darkfield condenser NA larger than the objective NA, DIC Wollaston prism splitting the beam into two orthogonally polarized beams whose phase difference is converted into intensity (fetch-tool summary).

Ques: Why does a high-magnification objective give a dimmer image?

Because brightness scales as $(NA/M{)}^2$ for transmitted bright field: light is spread over a larger magnified area, while collection rises only with $N{A}^2$ (my reading; TU Dresden for the $N{A}^2$ dependence). A 100x/0.95 objective is therefore about 10 times dimmer than a 10x/0.30 (computed), requiring longer exposure or brighter illumination. TU Dresden’s examples show why high-NA objectives exist at each magnification, 40x at 0.65 or 0.95: the higher NA gives more resolution and more light at the same magnification. Koehler illumination with the aperture diaphragm at about 70 to 80 per cent of the objective NA balances contrast and resolution (my reading; Dobbie for Koehler illumination). Reference: TU Dresden BioDIP, ‘The importance of Aperture in the Microscope’, teaching slides 06 Numerical Aperture (author not stated) (fetch-tool summary); I. Dobbie, ‘Brightfield techniques, contrast enhancement’, Micron lecture slides, University of Oxford: Koehler illumination separating imaging and illumination planes, darkfield condenser NA larger than the objective NA, DIC Wollaston prism splitting the beam into two orthogonally polarized beams whose phase difference is converted into intensity (fetch-tool summary).

References

1. KEYENCE, “Numerical Aperture (N.A.),” microscope glossary. Available: https://keyence.eu/ss/products/microscope/microscope_glossary/terminology/numerical_aperture.jsp
2. TU Dresden BioDIP, “The importance of Aperture in the Microscope” (06 Numerical Aperture), teaching slides. Available: https://biodip.webspace.tu-dresden.de/fileadmin/user_upload/Documents/Resources/Teaching_materials/theory_pdfs/06_Numerical-Aperture.pdf
3. Basler, “Microscopy resolution limits,” learning article. Available: https://www.baslerweb.com/en-in/learning/microscopy-resolution-limits/
4. Microspectra, “Microscope Formulas,” reference sheet. Available: https://www.microspectra.com/images/stories/files/Microscope Formulas(1).pdf
5. I. Dobbie, “Brightfield techniques, contrast enhancement,” Micron course lecture 5, University of Oxford, 2020. Available: https://downloads.micron.ox.ac.uk/lectures/micron_course_2020/lecture-05-brightfield-techniques.pdf

Related pages in this vault: Confocal Laser Scanning Microscope, White-Light Interferometer, Scanning Electron Microscope (SEM), Patterned-Wafer Optical Inspection, Emission Microscope, OBIRCH and Lock-In Thermography, Laser-Scattering Particle Inspection, Spectral Reflectometer, Overlay Metrology Tool, Raman Spectrometer, Calibration and Traceability.