Problem:

A glass rod of index n₂ = 1.5 has a hemispherical convex surface of radius R = 6 cm on one end as in in the figure below.  A ray makes an angle θ₁ equal to 0.1 radians with the plane PP at a height of x₁ = 1.6 cm from the optical axis.  The plane PP is 8 cm to the left of the vertex V.  Find the height of the ray x₂ and its direction θ₂ at the plane QQ 9 cm to the right of V in the paraxial approximation.

Solution:

• Reasoning:
  r₃’ = Mr₁.  M is the system matrix
  
• Details of the calculation:
  In this problem n₂ = 1.5, n₁ = 1, n₂ - n₁ = 0.5, R = 6 cm,
  z₂ - z₁ = 8 cm, z₃ - z₂ = 9 cm, x₁ = 1.6 cm, θ₁ = 0.1 radians.
  Therefore
  .
  Here all distances are measured in cm.
  x₃ = 1.8 cm, θ₂ = -0.1/1.5 = -0.067 radians.

Problem:

Find the image and object focal length of the above system.

Solution:

• Reasoning:
  The distances f’ = n₂/P and f = n₁/P are called the image and object focal length, respectively.
  P = R/(n₂ -n₁).
• Details of the calculation:
  The image focal length is f' = n₂R/(n₂ -n₁) = 18 cm
  and the object focal length is f = n₁R/(n₂ -n₁) = 12 cm.

Problem:

For an object located 18 cm to the left of V, find the image distance.

Solution:

• Reasoning:
  
• Details of the calculation:
  1/18 + 1.5/(z₃ - z₂) = 0.5/6,  (z₃ - z₂) = 54 cm.
  We have a real image.

Problem:

For an object located 6 cm to the left of V, find the image distance.

Solution:

• Reasoning:
  
• Details of the calculation:
  1/6 + 1.5/(z₃ - z₂) = 0.5/6,  (z₃ - z₂) = -18 cm.
  We have a virtual image.

Link:  Ray tracing with Excel

Please explore matrix multiplication in Excel.  You can find programs accomplishing the same tasks in Python (1interface) on Canvas, Modules, Numerical.