Maths for Optics

Question 1

what is the k-vector k?

Answer 1

k=2π/λ

Question 2

what is the wave number?

Answer 2

1/λ

Question 3

what is the angular frequency ω?

Answer 3

ω = 2π/τ

Question 4

what is the frequency ν?

Answer 4

ν = 1/τ

Question 5

What is the phase velocity?

Answer 5

wavelength/period

Question 6

what is the phase of a wave?

Answer 6

𝜙 = 𝑘𝑥 − 𝜔𝑡 − 𝜃

Question 7

what is the speed of a wave in terms of phase derivatives

Answer 7

v = − 𝜕𝜙/𝜕𝑡 / 𝜕𝜙/𝜕𝑥 = 𝜔/𝑘

Question 8

What is the complex form of a wave?

Answer 8

Ḛ (𝑥, 𝑡) = Ḛ0 exp 𝑖(𝑘𝑥 − 𝜔𝑡)

Question 9

what is the Kronecker delta function?

Answer 9

1 if m=n
0 if m does not = n

Question 10

What is the Fourier Transform formula?

Answer 10

𝐹(𝜔) = integral from −∞ to ∞ of 𝑓(𝑡) exp(−𝑖𝜔𝑡)𝑑𝑡

Question 11

What is the inverse fourier transform function?

Answer 11

f(t) = 1/2𝜋 the integral from −∞ to ∞ of F(𝜔) exp(−𝑖𝜔𝑡)𝑑𝜔

Question 12

What do we define as the spectrum of a wave?

Answer 12

|ℱ {𝐸(t)}|^2

Question 13

What is the fourier transform of a rect(t) function

Answer 13

sinc(𝜔/2)

Question 14

what is the fourier transform of a decaying exponential exp(-at)

Answer 14

-i/𝜔-ia

Question 15

what is the fourier transform of a gaussian exp(-at^2)?

Answer 15

exp(-𝜔^2/4a)

Question 16

what is the dirac delta function?

Answer 16

∞ if 𝑡 = 0
0 if 𝑡 ≠ 0

Question 17

what is the integral of the dirac delta between ∞ and -∞

Answer 17

1

Question 18

What is the function exp(i𝜔0t)

Answer 18

it is a pure frequency (fourier transforms to a dirac delta)

Question 19

What is the fourier transform of a scaled function?

Answer 19

ℱ {𝑓 (𝑎𝑡)} = 𝐹((𝜔/𝑎)/|𝑎|

Question 20

what is the convolution of f and g?

Answer 20

𝑓 (𝑡) ⨂𝑔 (𝑡) = integral from −∞ to ∞ of 𝑓(𝜏) 𝑔 (𝑡-𝜏)𝑑𝜏

Question 21

State the convolution theorem

Answer 21

The fourier transform of the convolution of two functions is given by the product of the fourier transforms of the individual functions