We now use our matrix formalism from WS1. This is the formalism adopted by Siegman (a recommended text). For this, a ray at any plane, i, is defined as: r₁ = [(~), where x is the lateral position and x' = na is the index-modified ray angle. We approximate n = 1 for air, and so x' = α in air. Ref index, n X a Optical axis [A In transferring a ray from a plane 1 to a plane 2: [*]₂ = [ĉ B], []₁ = M21 | [X] ལ། ГА В In transferring a ray from a plane a plane 2 to a plane 3: [✓] = [^_B]¸₂[]₂ = ' = M32 [X] It follows that to transfer the ray from plane 1 to plane 3 is given by the product: [*]3 = M 32M21 | and so: M31 =M32M 21. Note the order of the matrices is from right to left in terms of the ordering of light interactions. This ordering of matrices remains the same irrespective of the direction of a light vector, and so even though light generally goes from left to right in our illustrations, these two things shouldn't be muddled. Keeping to our formalism for the numeric labelling of the matrices helps as a check of their ordering. With this formalism, only the first and last number is kept for the equivalent system matrix. Use the following matrices in answering the following questions i) A thin lens: MI=/f where f is the lens focal length. ii) Free propagation: Mp = [1 d/n], approximate n = 1 for air. where d is the displacement and n is the refractive index. We Question 1 Illustrated is a thin lens placed in air. 1 Front side of lens Back side of lens Optical axis SI a b S2 2 (a) By taking a product of three matrices, determine the system transfer matrix, M₁ = M2b.mba .Ma1, that transfers rays from a plane 1 to a plane 2. Take care to define variables used in each matrix. [1]

We now use our matrix formalism from WS1. This is the formalism adopted by Siegman (a recommended text). For this, a ray at any plane, i, is defined as: r₁ = [(~), where x is the lateral position and x' = na is the index-modified ray angle. We approximate n = 1 for air, and so x' = α in air. Ref index, n X a Optical axis [A In transferring a ray from a plane 1 to a plane 2: []₂ = [ĉ B], []₁ = M21 | [X] ལ། ГА В In transferring a ray from a plane a plane 2 to a plane 3: [✓] = [^_B]¸₂[]₂ = ' = M32 [X] It follows that to transfer the ray from plane 1 to plane 3 is given by the product: []3 = M 32M21 | and so: M31 =M32M 21. Note the order of the matrices is from right to left in terms of the ordering of light interactions. This ordering of matrices remains the same irrespective of the direction of a light vector, and so even though light generally goes from left to right in our illustrations, these two things shouldn't be muddled. Keeping to our formalism for the numeric labelling of the matrices helps as a check of their ordering. With this formalism, only the first and last number is kept for the equivalent system matrix. Use the following matrices in answering the following questions i) A thin lens: MI=/f where f is the lens focal length. ii) Free propagation: Mp = [1 d/n], approximate n = 1 for air. where d is the displacement and n is the refractive index. We Question 1 Illustrated is a thin lens placed in air. 1 Front side of lens Back side of lens Optical axis SI a b S2 2 (a) By taking a product of three matrices, determine the system transfer matrix, M₁ = M2b.mba .Ma1, that transfers rays from a plane 1 to a plane 2. Take care to define variables used in each matrix. [1]