Matrix Methods for Optics

Question 1

What does the matrix method allow us to achieve and how?

Answer 1

Using the paraxial approximation and matrices, we can analyse changes in ray height and direction

Question 2

What is the paraxial approximation?

Answer 2

tan 𝛼0 ≅ 𝛼0

Question 3

What is the matrix form of the translation matrix?

Answer 3

y1 = |1 L | |yo|
𝛼1 = |0 1 | |𝛼0|
y is the height and alpha is the angle

Question 4

What is the refraction matrix?

Answer 4

y’ = | 1 0| |y|
𝛼’ = |(1/R)((n/n’)-1) n/n’ | |𝛼|

Question 5

what is the reflection matrix?

Answer 5

y’ = |1 0 | |y|
𝛼’ = |2/R 1 | |𝛼|

Question 6

What order do matrices operate on light?

Answer 6

the order in which the optical interfaces do

Question 7

What is the whole system ray transfer matrix?

Answer 7

M = MNMN-1…M2M1

Question 8

what is the thin lens matrix?

Answer 8

M = | 1 0|
|-1/f 1|

Question 9

What is the form of the transfer matrix?

Answer 9

M = |A B|
|C D|

Question 10

What is the determinant of the transfer matrix in terms of the starting and ending refractive indices?

Answer 10

Det = AD-BC = no/nt

Question 11

What are the linear equations from the ABCD matrix?

Answer 11

yf = Ayo + B𝛼0
𝛼𝑓 = Cy0 + D𝛼0

Question 12

What happens if D = 0 in the ABCD matrix?

Answer 12

𝛼𝑓 = 𝐶𝑦0 so is independent of the input angle
input plane coincides with the front focal plane

Question 13

What happens if A = 0 in the ABCD matrix?

Answer 13

yf = B𝛼0 and the output rays are independent of the input height and all converge at the same output height
they form at the back focal plane

Question 14

What happens if B = 0 in the ABCD matrix?

Answer 14

yf = Ay0 implying conjugate points
A will be the linear magnification

Question 15

What happens if C=0 in the ABCD matrix?

Answer 15

𝛼𝑓 = 𝐷𝛼0 indicating parallel rays coming in produce parallel rays coming out
D is the angular magnification

Question 16

What is the matric for refraction at a plane surface

Answer 16

yf = |1 0||y0|
𝛼𝑓 = |0 n1/n2||𝛼0|