Introduction to Geometrical Optics:
Sample problems solutions supplementary slides
Apratim Majumder
Optics for Energy 2020
2020/10/13
Fish (Wanda) is the input side
Olive (person) is the output
side
Light ray travels from right to left
Fish (Wanda) is the input side
Olive (person) is the output
side
Light ray travels from right to left
𝑛𝑜𝑢𝑡𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝑖𝑛𝛼𝑖𝑛𝑥𝑖𝑛
From Output side to Input side
Fish (Wanda) is the input side
Olive (person) is the output
side
Light ray travels from right to left
𝑛𝑜𝑢𝑡𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝑖𝑛𝛼𝑖𝑛𝑥𝑖𝑛
Here, 𝑛𝑜𝑢𝑡 = 1 and 𝑛𝑖𝑛 = 𝑛Hence:
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
From Output side to Input side
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 −1−𝑛
−𝑅
0 1
1 0𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 −1−𝑛
−𝑅
0 1
1 0𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 +1−𝑛
𝑅
𝑅
𝑛
1−𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 −1−𝑛
−𝑅
0 1
1 0𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 +1−𝑛
𝑅
𝑅
𝑛
1−𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1
𝑛
1 − 𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 −1−𝑛
−𝑅
0 1
1 0𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 +1−𝑛
𝑅
𝑅
𝑛
1−𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1
𝑛
1 − 𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1
𝑛
1 − 𝑛
𝑅𝑠
𝑛+𝑅
𝑛𝑠1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation air
(𝑠)
Refractionat
spherical surface (R)
Free spacepropagation inside water
(𝑠′)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 −1−𝑛
−𝑅
0 1
1 0𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1 +1−𝑛
𝑅
𝑅
𝑛
1−𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠 1
1
𝑛
1 − 𝑛
𝑅𝑅
𝑛1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1
𝑛
1 − 𝑛
𝑅𝑠
𝑛+𝑅
𝑛𝑠1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
Imaging Condition: M21 = 0
𝑠
𝑛+𝑅
𝑛= 0
𝑠 = −𝑅
Fish (Wanda) is the output
side
Olive (person) is the input
side
Light ray travels from left to right
Fish (Wanda) is the output
side
Olive (person) is the input
side
Light ray travels from left to right
𝑛𝑜𝑢𝑡𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝑛𝑖𝑛𝛼𝑖𝑛𝑥𝑖𝑛
From Output side to Input side
Fish (Wanda) is the output
side
Olive (person) is the input
side
Light ray travels from left to right
𝑛𝑜𝑢𝑡𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝑛𝑖𝑛𝛼𝑖𝑛𝑥𝑖𝑛
Here, 𝑛𝑜𝑢𝑡 = 𝑛 and 𝑛𝑖𝑛 = 1Hence:
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
From Output side to Input side
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛1 + 𝑠
1 − 𝑛
𝑅+ 𝑠
𝑠′
𝑛
1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛1 + 𝑠
1 − 𝑛
𝑅+ 𝑠
𝑠′
𝑛
1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 + 𝑠
1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛+ 𝑠 +
𝑠𝑠′
𝑛𝑅(1 − 𝑛) 1 +
𝑠′
𝑛𝑅(1 − 𝑛)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛1 + 𝑠
1 − 𝑛
𝑅+ 𝑠
𝑠′
𝑛
1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 + 𝑠
1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛+ 𝑠 +
𝑠𝑠′
𝑛𝑅(1 − 𝑛) 1 +
𝑠′
𝑛𝑅(1 − 𝑛)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
Free spacepropagation inside water
(𝑠′)
Refractionat
spherical surface (R)
Free spacepropagation in air
(𝑠)
𝛼𝑖𝑛𝑥𝑖𝑛
1 0𝐷
𝑛1
Free space propagation
matrix
1 −𝑛𝑜𝑢𝑡 − 𝑛𝑖𝑛
𝑅0 1
Refraction at spherical surface
matrix
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 −𝑛−1
𝑅
0 1
1 0𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
(since 𝑠′ = 𝑅)
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 0𝑠′
𝑛1
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=
1 + 𝑠1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛1 + 𝑠
1 − 𝑛
𝑅+ 𝑠
𝑠′
𝑛
1 − 𝑛
𝑅+ 1
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
𝑛𝛼𝑜𝑢𝑡𝑥𝑜𝑢𝑡
=1 + 𝑠
1 − 𝑛
𝑅
1 − 𝑛
𝑅𝑠′
𝑛+ 𝑠 +
𝑠𝑠′
𝑛𝑅(1 − 𝑛) 1 +
𝑠′
𝑛𝑅(1 − 𝑛)
𝑛𝛼𝑖𝑛𝑥𝑖𝑛
Imaging Condition: M21 = 0
𝑠′
𝑛+ 𝑠 +
𝑠𝑠′
𝑛𝑅(1 − 𝑛) = 0