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How to Use the Matrix Approach to Calculate Cardinal Points

Cardinal points are used to describe the behavior of an optical system, and to estimate the location and size of an image. These six points are the two focal points, the two principal points, and the two nodal points. The focal points are the locations of a viewed object, and the location of its image beyond the lens. The principal points are the points where the principal planes cross the optical axis. The nodal points are where an entering ray of light and an exiting ray cross the optical axis. Along with the focal lengths, the locations of the cardinal points can be determined using the matrix of the optic system.

Things You'll Need

Calculator

Instructions

- 1
  Set up the two-by-two system matrix. The top two terms, A and B' are coefficients for the equation denoting the point at which light from the image hits the lens. The bottom two terms, C and D, are coefficients for the equation determining the angle between the horizontal axis and the ray of light once it enters the lens.
- 2
  Calculate the focal points. The first focal point is D/C, and the second is ̵1;A/C.
- 3
  Calculate the principal points. The first principal point is (D̵1;nʹ/nʺ)/C, where nʹ is the refraction index of the incoming ray and nʺ is the refraction index of the outgoing ray. When the refraction indices are identical, the first principal point is (D-1)/C. The second principal point is (1-A)/C.
- 4
  Calculate the nodal points. The first nodal point is (D-1)/C. The second nodal point is (nʹ/nʺ-A)/C, or (1-A)/C when the incoming and outgoing refraction indices coincide. When nʹ = nʺ, the principal points are identical to the nodal points. As light enters the lens, it is refracted at the first nodal point, then refracted again at the second nodal point.
- 5
  Calculate the focal lengths. The first focal length is (nʹ/nʺ)/C, and the second focal length is 1/C. When the refraction indices are the same, both focal lengths are 1/C. Focal lengths are the distances between the focal points and their corresponding principal points.

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