Links

• Field of View

  • https://www.microscopyu.com/microscopy-basics/field-of-view
  • http://www.dynamicscience.com.au/tester/solutions1/biology/microscopy/fieldofview.html

• Resolution and Numerical Aperture

  • https://www.microscopyu.com/microscopy-basics/resolution
  • https://micro.magnet.fsu.edu/primer/anatomy/numaperture.html
  • http://zeiss-campus.magnet.fsu.edu/articles/basics/resolution.html

• Chromatic Aberration

  • https://www.microscopyu.com/tutorials/chromatic

• Working Distance

  • https://www.microscopyu.com/microscopy-basics/working-distance-and-parfocal-length

Intro

• Your first assignment as a research assistant in a new lab is to image and document sections of a mouse embryo.

• Because the embryo is small, the animal has been sectioned intact along the sagittal plane.

• Other workers in the lab have already prepared the samples for you.

• Each section of the embryo is 250 μm thick.

• These sections were then stained, mounted on standard microscope slides, and cover slipped.

• Before beginning this task, you will need to be aware of the strengths and limitations of the microscopy equipment you have on hand.

• With this information you can then make appropriate decisions on how to image these samples to efficiently obtain the relevant data from these slides.

• You should refer to this online viewer to assist you in working through these decisions:

  • http://www.emouseatlas.org/eAtlasViewer_ema/application/ema/kaufman/plate_31a.php

  

  • Let Image Width = $17.15mm$$17.15\ mm$

  • Let Image Height = $28.3mm$$28.3\ mm$

    • $\therefore$$\therefore$ Image Area = $17.15mm\ast 28.3mm=485.345m{m}^2$$17.15\ mm * 28.3\ mm = 485.345\ mm^2$

Notes

• Objective = series of lenses that keeps image flat and colors constant

• You are always limited by what you can see by field number of eye piece

  • eye piece always has the smallest field number compared to objective lens

• Rectangular viewports have independent fields of view for length and width.
• Whereas for the eyepiece , we can just use the area formula for a circle

• Resolution = "distance between 2 points where you can still tell there are 2 objects"

• "you can buy a very low resolution optic , with a really high $NA$$NA$"

• "we need to see at least 3 microns or 3000 nano meters"

  • "so we need a resolution of < 3000 nm"

Topic 1: Field of View

• How big this embryo section?

  • (zoom out; right click on image to access measurement mode; pick two points)

• Come up with answer in mm²

• How much of the embryo will be visible through the microscope at a time?

• What is the field number ( FN ) of the different objectives available on the microscope?

• This is an image of the eye piece on your microscope.

  • Magnification = $10\text{x}$$10\ \text{x}$
  • Field Number = $22$$22$

• If the objective and eye piece field numbers do not match ,

  • your image will be limited by the smaller of the two numbers

• Calculate the field of view ( in mm ) for the following objectives :

  • $10\text{x}$$10 \text{x}$ = $\frac{22}{10\text{x}}=2.2$$\frac{22}{10 \text{x}} = 2.2$
  • $20\text{x}$$20 \text{x}$ = $\frac{22}{20\text{x}}=1.1$$\frac{22}{20 \text{x}} = 1.1$
  • $40\text{x}$$40 \text{x}$ = $\frac{22}{40\text{x}}=0.55$$\frac{22}{40 \text{x}} = 0.55$
  • $60\text{x}$$60 \text{x}$ = $\frac{22}{60\text{x}}=0.36$$\frac{22}{60 \text{x}} = 0.36$

• The FOV represents the diameter of the circle visible through the eyepieces.

• Calculate the visible area ( in mm² ) and percent of total area of the embryo it represents for the following objectives :

  • $10\text{x}$$10\ \text{x}$ :
  $$\text{Visible Area}=\pi \ast {\left(\frac{\frac{22}{10\text{x}}}{2}\right)}^2=3.8013271108436504m{m}^2$$
  $$\text{Percent of Total Area}=\frac{3.8013271108436504m{m}^2}{485.345m{m}^2}\ast 100\mathrm{\%}=0.7832216486918894\mathrm{\%}$$

  • $60\text{x}$$60\ \text{x}$ :

    $$\text{Visible Area}=\pi \ast {\left(\frac{\frac{22}{60\text{x}}}{2}\right)}^2=0.10559241974565692m{m}^2$$
    $$\text{Percent of Total Area}=\frac{0.10559241974565692m{m}^2}{485.345m{m}^2}\ast 100\mathrm{\%}=0.021756156908108032\mathrm{\%}$$

• The digital cameras attached to microscopes have square or rectangular sensors

  • so they won’t capture everything you can see by eye through the microscope.

• Your lab microscope is equipped with a SPOT RT3 camera.

• Using the specification sheet, determine the effective field number ( FN ) of this camera.

• What percent of the embryo will be captured in each image you would take using a 10x and 60x objective?

  • $10\text{x}$$10\ \text{x}$ :
  $$\text{Visible Area}={\left(\frac{11.8mm}{10\text{x}}\right)}_{\text{FOV-length}}\ast {\left(\frac{8.9mm}{10\text{x}}\right)}_{\text{FOV-width}}=1.0502m{m}^2$$
  $$\text{Percent of Total Area}=\frac{1.0502m{m}^2}{485.345m{m}^2}\ast 100\mathrm{\%}=0.21637814873267766\mathrm{\%}$$
  • $60\text{x}$$60\ \text{x}$ :
  $$\text{Visible Area}={\left(\frac{11.8mm}{60\text{x}}\right)}_{\text{FOV-length}}\ast {\left(\frac{8.9mm}{60\text{x}}\right)}_{\text{FOV-width}}=0.029172m{m}^2$$
  $$\text{Percent of Total Area}=\frac{0.029172m{m}^2}{485.345m{m}^2}\ast 100\mathrm{\%}=0.006010504131463268\mathrm{\%}$$

• How many images will be necessary to document the whole embryo?

  • $10\text{x}$$10\ \text{x}$ :
  $$\text{Number of Images Required}=\frac{485.345m{m}^2}{1.0502m{m}^2}=462.1453056560654$$
  • $60\text{x}$$60\ \text{x}$ :

Topic 2: Resolution and Numerical Aperture

• Rayleigh Equations :

  • https://en.wikipedia.org/wiki/Airy_disk#Mathematical_formulation
  • https://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
  • https://physics.stackexchange.com/questions/264925/where-does-the-0-61-come-from-in-r-sin-theta-0-61-lambda
  • https://byjus.com/physics/rayleigh-criterion/
  • http://kmdouglass.github.io/posts/simple-pupil-function-calculations/

  

• Where :

  • $\alpha$$\alpha$ = radius of the aperture

  • $\theta$$\theta$ = angle of observation

    • ie , the angle between the axis of the circular aperture and the line between aperture center and observation point

• Where :

  • $\alpha$$\alpha$ = radius of the aperture
  • $q$$q$ = radial distnce from observation point to the optical axis
  • $R$$R$ = distance to the aperture

• https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind:_J%CE%B1
• Let alpha ( $\alpha$$\alpha$ ) = 1

• Where :

  • $\mathrm{\Gamma }$$\Gamma$ = gamma function

• Solving for first zero ( Wolfram Alpha ):

  $$x\approx \pm 3.83170597020751231561443588630816076656454527428780192876229898991883930951901147021411287475742312672447473309723985960770829353934295074637568330729735079473543886122516050913293101$$

• Let Wave Number ( $k$$k$ ) = $\frac{2\ast \pi }{\lambda }$$\frac{2 * \pi}{\lambda}$

  • Where $\lambda$$\lambda$ = wavelength of light

• Assume $n=1$$n = 1$ ( refractive index of air )

• Isolating for $sin(\theta )$$sin(\theta)$ :

  • Division of 2π

• Putting everything back in terms of resolution via the Rayleigh equation :

• While collecting images with the 60X objective will require significantly more time , it may be necessary depending on the level of detail you wish to capture.

  • This means you’ll need to know something about the resolving power of your microscope.

• Zoom in again on the embryo online at emouseatlas.org.

• What is the size of some of the smallest objects or features in the image?

  • $20\mu m$$20\ \mu m$

• What kind of resolution would you need to achieve to replicate this level of detail?

  • We need the resolution to be smaller than $20\mu m$$20\ \mu m$
  $$\frac{{\lambda }_x}{2\ast {\text{NA}}_y}<(20\mu m{)}_{\text{resolution}}$$

  • Solving for ${\lambda }_x$$\lambda_x$ :

    $$\begin{array}{}\text{( where y ≠ 0 )}& {\lambda }_x<40\mu m\ast {\text{NA}}_y\end{array}$$

  • Solving for ${\text{NA}}_y$$\text{NA}_y$ :

  $$\begin{array}{}\text{( where x ≠ 0 )}& {\text{NA}}_y<\frac{{\lambda }_x}{40\mu m}\end{array}$$

• Using light at 550nm , calculate the resolution possible using objectives with the following numerical apertures ( NA ).

  • $\text{NA}=0.2$$\text{NA} = 0.2$ :
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.2}=1375.0nm$$
  • $\text{NA}=0.3$$\text{NA} = 0.3$
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.3}=916.667nm$$
  • $\text{NA}=0.4$$\text{NA} = 0.4$ :
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.4}=687.5nm$$
  • $\text{NA}=0.95$$\text{NA} = 0.95$ :
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.95}=289.47368421052636nm$$
  • $\text{NA}=1.35$$\text{NA} = 1.35$ :
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 1.35}=203.7037037037037nm$$
  • $\text{NA}=1.4$$\text{NA} = 1.4$ :
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.95}=196.42857142857144nm$$

• Review the NA of the objectives available on the lab microscope.

• Do your calculations influence or limit your choice of objectives? Why or why not?

  • UPLFLN 10X2 :

    • W.D. = 10.0 , MAG = 10x , F.N. = 26.5 , NA = 0.3 , IM = none
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.3}=916.667nm\ast \frac{1\mu m}{1000nm}=0.91667\mu m<20\mu m$$

  • UPLSAPO 10X2 :

    • W.D. = 3.1 , MAG = 10x , F.N. = 26.5 , NA = 0.4 , IM = none
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.4}=687.5nm\ast \frac{1\mu m}{1000nm}=0.6875\mu m<20\mu m$$

  • UPLSAPO 20X :

    • W.D. = 0.6 , MAG = 20x , F.N. = 26.5 , NA = 0.75 , IM = none
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.75}=366.667nm\ast \frac{1\mu m}{1000nm}=0.3667\mu m<20\mu m$$

  • UPLSAPO 40X2 :

    • W.D. = 0.18 , MAG = 40x , F.N. = 26.5 , NA = 0.95 , IM = none
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 0.95}=289.4737nm\ast \frac{1\mu m}{1000nm}=0.2894\mu m<20\mu m$$

  • UPLSAPO 60XO :

    • W.D. = 0.15 , MAG = 60x , F.N. = 26.5 , NA = 1.35 , IM = oil ( n = 1.515 )
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 1.35}=203.7037037nm\ast \frac{1\mu m}{1000nm}=0.2037037\mu m<20\mu m$$

  • UPLSAPO 100XS :

    • W.D. = 0.2 , MAG = 100x , F.N. = 22.0 , NA = 1.35 , IM = silicon oil ( n = 1.45 to 1.50 ( source ) )
  $$\text{Resolution ( r )}=\frac{550nm}{2\ast 1.35}=203.7037037nm\ast \frac{1\mu m}{1000nm}=0.2037037\mu m<20\mu m$$

  • SPOT^TM RT3 2.0 MP Slider :

    • FOV = 11.8 mm x 8.9 mm

• The wavelength of light is a key parameter in the equation for resolution.

• This suggests the choice of wavelength influences the resolution possible for your microscope.

• Calculate the maximum resolution possible with a high numerical aperture ( 1.40 ) objective.

  • $\text{Wavelength}(\lambda )=400nm$$\text{Wavelength}\ (\ \lambda\ ) = 400\ nm$ :
  $$\text{Resolution ( r )}=\frac{400nm}{2\ast 1.40}=142.85714285714286nm$$
  • $\text{Wavelength}(\lambda )=500nm$$\text{Wavelength}\ (\ \lambda\ ) = 500\ nm$ :
  $$\text{Resolution ( r )}=\frac{500nm}{2\ast 1.40}=178.57142857142858nm$$
  • $\text{Wavelength}(\lambda )=600nm$$\text{Wavelength}\ (\ \lambda\ ) = 600\ nm$ :
  $$\text{Resolution ( r )}=\frac{600nm}{2\ast 1.40}=214.2857142857143nm$$
  • $\text{Wavelength}(\lambda )=700nm$$\text{Wavelength}\ (\ \lambda\ ) = 700\ nm$ :
  $$\text{Resolution ( r )}=\frac{700nm}{2\ast 1.40}=250.00nm$$

• Using the equation for NA ( $\text{NA}=n\ast sin(\alpha )$$\text{NA} = n * sin(\ \alpha\ )$ ) , determine the maximum theoretical NA for a dry ( air ) objective.

  • let refractive index of air = 1.00
  $$\text{NA}=1.0\ast sin({90}^{\circ })=1.0$$

• Explain why.

  

  • Max Half-Angle light can enter into objective is ${90}^{\circ }$$90^{\circ}$

Topic 3: Aberrations

• Distance from the lense where everything comes into focus.

• Big lens with smooth surface has difference than small lense with large curvature.

• Different wavelengths as they pass thorugh the material have different refractive indices.

  • Different wavelengths converge at different focal points

• Chromatically corrected lens focuses everything at the correct place

• Fluorite = cheapest lens

• Aberration corrections and numerical drive the price of the lens

• What kind of staining is used on this embryo tissue?

  • Stain = haematoxylin and eosin

    • https://academic.oup.com/gigascience/article/7/2/gix131/4768197

• Use the simplified lens maker formula to calculate the changes in focal length of a lens for the wavelengths listed

• Interactive Lens-Marker's Equation :

  • https://39363.org/NOTES/WSU/2022/Spring/NEU3400/MISC/Lens-Maker-Equation.html
  $$\frac{1}{F}=(n-1)\ast (\frac{1}{R_1}-\frac{1}{R_2})$$
  $$F=\frac{1}{(n-1)\ast (\frac{1}{R_1}-\frac{1}{R_2})}$$

  

   

  • Let $(\frac{1}{R_1}-\frac{1}{R_2})=1$$\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = 1$

     

  • Red light ( n = 1.513 ) :

  $$F=\frac{1}{(1.513-1)\ast \left(1\right)}=1.9493177387914233$$
  • Violet light ( n = 1.532 ) :
  $$F=\frac{1}{(1.532-1)\ast \left(1\right)}=1.8796992481203008$$

• Will these differences affect your imaging?

  $$F\frac{1}{\propto }n$$
  • Violet focuses in front of the focal point
  • Red focuses further away from the focal point than violet light
  • Spherical aberration , anything on the edges gets distorted

Topic 4: Working Distance

• Every objective has a limited working distance.

• You must keep this in mind when choosing objectives for your studies.

• In order to safely focus on any region of your sample ,

  • the sum of the tissue sample thickness along with the coverslip thickness must be less than the working distance of the objective.

• Your coworker who prepared the embryo sections informs you that $1.5$$1.5$ coverslips were used to mount the sections.

• Working Distance = Coverslip thickness + distance to objective

• Coverslip types and thickness tolerances :

  • $\text{No. 0}=0.08-0.12mm$$\text{No. 0} = 0.08\ -\ 0.12\ mm$

  • $\text{No. 1}=0.13-0.16mm$$\text{No. 1} = 0.13\ -\ 0.16\ mm$

  • $\text{No. 1.5}=0.16-0.19mm$$\text{No. 1.5} = 0.16\ -\ 0.19\ mm$

    • ( $0.17mm$$0.17\ mm$ = typical )

  • $\text{No. 0}=0.19-0.23mm$$\text{No. 0} = 0.19\ -\ 0.23\ mm$

• Review the spec sheets for the objective on your microscope.

• So we need a working distance bigger than 0.42 mm

• Are there any limitations on which objectives can be used to image the embryo sections?

  • UPLSAPO 40X2 , UPLSAPO 60XO , UPLSAPO 100XS have working distances smaller than $0.42mm$$0.42\ mm$

    • so they will NOT work with our cover slip and tissue thickness

• Explain why.

  • Only UPLFLN 10X2 , UPLSAPO 10X2 , UPLSAPO 20X lenses have a working distance larger than $0.42mm$$0.42\ mm$
  • as you get higher magnification , the working distance goes down

• Review the spec sheets for the objectives available on your microscope.

• Is there a relationship between NA and working distance? Explain why or why not.

  • https://en.wikipedia.org/wiki/Numerical_aperture

    • Increasing the magnification and the numerical aperture of the objective reduces the working distance

  • https://physics.stackexchange.com/a/528093

    • For a given numerical aperture , one can usually increase the working distance

      • but it requires making all the lens elements larger

    • For a given numerical aperture , there is a tradeoff between working distance and cost

      • because larger elements will cost more.
  $$\text{Numerical Aperature (NA)}\frac{1}{\propto }\text{Resolution}$$
  $$\text{Magnification}\frac{1}{\propto }\text{Working Distance}$$
  $$\text{NA}\propto \text{’best resolution’}$$

• Lastly , summarize your strategy for documenting these embryo sections.

• What objective( s ) are best?

  • UPLSAPO 10X2

• Why did you choose them?

  • want to be able to focus through the whole specimen , and take as little images as possible
  • UPLSAPO 10X2 has higher NA so resolution is lower ( "better resolution" )
  • apo corrected