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How to Calculate Dimensions of Focal Volume

Optics is a simple science. Light operates according to straightforward rules and it interacts with materials and changes. The complexity of optics arises because of the infinite variety possible in the interactions of light and matter and the number of properties of light that can be manipulated. To deal with complexity, scientists have developed a number of concepts to simplify their work. One of those concepts is called the focal volume, and it's pretty easy to calculate. You need to pick the diameter that corresponds to your requirements, something like the size of a single detector pixel, or the diameter wherein a laser is bright enough to make a fluorophore shine.

Instructions

- 1
  Calculate the F-number (often written "F/#") of your system by dividing the diameter by the focal length. As an example, assume you've got a 100 mm focal length lens with a 40 mm diameter. The F/# is 100/40 = 2.5, which is often written as F/2.5.
- 2
  Calculate the diffraction-limited spot size of your system as 2.44 x F/# x wavelength. For visible light, you can take the wavelength to be 550 nanometers, which is 550 x 10^-6 millimeters. So the example system has a diffraction-limited spot size of
  
  2.44 x 2.5 x 550 x 10^-6 = 3.4 x 10^-3 mm.
- 3
  Find the angle at which the lens bends the light. The light coming from one edge of the lens is bent at an angle to cross the optical axis of the lens at a distance of one focal length away. Mathematically, that behavior is expressed by the equation
  
  tangent (angle) = diameter/2/focal length = 1/(2 x F/#).
- 4
  Calculate the length of the triangle with an angle defined by the previous step and the side equal to the radius of the diffraction-limited spot. The length of that triangle is given by
  
  length = (1.22 x lambda x F/#)/tan (angle); substituting the earlier expression for tan (angle)
  
  length = (1.22 x lambda x F/#)/(1/(2 x F/#))
  
  length = (2.44 x lambda x (F/#)^2).
- 5
  Multiply the area of the spot you chose by 2 times the length of the triangle calculated in the previous step. For the example problem, in equation form, this is
  
  volume = pi x (1.22 x lambda x F/#)^2 x 2 x (2.44 x lambda x (F/#)^2)
  
  volume = pi x 1.22^2 x 2 x 2.44 x (550 x 10^-6)^3 x (2.5)^4
  
  volume = 1.5 x 10^-7 mm^3.

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