Optics: Equations

Question 1

What is Maxwell’s equation for the (charged) source of an electric field in a vacuum?

Answer 1

∇ = del operator
E = electric field

Question 2

What is Maxwell’s equation for the (charged) source of a magnetic field in a vacuum?

Answer 2

∇ = del operator
B = magnetic field

Question 3

What is Maxwell’s equation for electromagnetic induction (Faraday’s Law)?

Answer 3

∇ = del operator
E = electric field
B = magnetic field

Question 4

What is Maxwell’s equation for Ampere’s Law?

Answer 4

∇ = del operator
B = magnetic field
E = electric field
ε₀ = dielectric constant in a vacuum
µ₀ = permeability in a vacuum
c = speed of light

Question 5

What is the wave equation for an electric field?

Answer 5

∇ = del operator
E = electric field
c = speed of light

Question 6

What is the wave equation for a magnetic field?

Answer 6

∇ = del operator
B = magnetic field
c = speed of light

Question 7

What is the equation for the wavelength in a vacuum?

Answer 7

λ₀ = wavelength in a vacuum
c₀ = speed of light in a vacuum
ν = frequency

Question 8

What is the equation for the wavelength in a medium?

Answer 8

λ’ = wavelength in a medium
c’ = speed of light in a medium
ν = frequency

Question 9

What is the solution to the wave equation in one dimension?

Answer 9

E₀ = amplitude (could also be B₀ for a magnetic wave)
k = wavenumber = 2π/λ
ω = angular frequency = 2πν

Question 10

What is the solution to the wave equation in three dimensions?

Answer 10

E₀ = amplitude
k = wave vector
ω = angular frequency = 2πν

Question 11

What is the solution of the wave equation for a plane wave?

Answer 11

E₀ = amplitude
r = (x, y, z)
k = wave vector
ω = angular frequency = 2πν
c.c = complex conjugate

Question 12

What is the solution of the wave equation for a spherical wave?

Answer 12

E₀ = amplitude
r = |r| = √(x² + y² + z²)
k = wave vector
ω = angular frequency = 2πν
c.c = complex conjugate

Question 13

What is the equation for the Poynting vector?

Answer 13

S = Poynting vector
E = electric field
B = magnetic field
µ₀ = permeability in a vacuum

Question 14

What is the equation for the index of refraction?

Answer 14

c = speed of light in a vacuum
c’ = speed of light in a medium
εᵣ = ε/ε₀ = dielectric constant in the medium
µᵣ = µ/µ₀ = permeability of the medium

Question 15

What is the equation for Snell’s Law?

Answer 15

n₁ = refractive index of medium 1
n₂ = refractive index of medium 2
θ₁ = angle of incidence
θ₂ = angle of refraction

Question 16

What is the equation for the critical angle?

Answer 16

Question 17

What is the equation for Fermat’s Principle?

Answer 17

t = time
d = distance
nd = L = optical path length (n = local refractive index)
c’ = speed in a medium
c = speed in a vacuum

Question 18

What is the equation used to calculate the optical path length?

Answer 18

The minimal value of this equation is found.

L(r) = optical path length
n(r) = refractive index
A = initial point
B = final point

Question 19

What is the imaging equation?

Answer 19

s = distance from object to optical element
s’ = distance from image to optical element
f = r/2 = focal length

Question 20

What is the equation for magnification?

Answer 20

M = magnification
h’ = image height
h = object height
s’ = distance from image to optical element
s = distance from object to optical element

Question 21

What is the imaging equation for image formation at a spherical surface?

Answer 21

n₁, n₂ = refractive index
s = distance from object to optical element
s’ = distance from image to optical element

Question 22

What is the magnification equation for a spherical refractive surface?

Answer 22

M = magnification
h’ = image height
h = object height
n₁, n₂ = refractive index
s’ = distance from image to optical element
s = distance from object to optical element

Question 23

What is the focal length of a thin lens that is curved on both sides?

Answer 23

f = total focal length
f₁ = focal length (side 1)
f₂ = focal length (side 2)

Question 24

What is the imaging equation for a thin lens in a medium?

Answer 24

n₁ = refractive index of the medium
n₂ = refractive index of the lens
r₁ = radius of lens (side 1)
r₂ = radius of lens (side 2)
s = distance from object to optical element
s’ = distance from image to optical element
f = focal length

Question 25

What is Lensmaker’s equation?

Answer 25

n₁ = refractive index of the medium n₂ = refractive index of the lens r₁ = radius of lens (side 1) r₂ = radius of lens (side 2) f = focal length

Question 26

What is the magnification equation for a thin lens?

Answer 26

m = magnification m₁ = magnification of lens (side 1) m₂ = magnification of lens (side 2) n₁ = refractive index of the medium n₂ = refractive index of the lens s = distance from object to optical element s’ = distance from image to optical element

Question 27

What is the equation for the visual angle (or angular diameter / apparent size) of an object?

Answer 27

h = object height ε₀ = visual angle

Question 28

What is the equation for the (angular) magnification for an optical instrument?

Answer 28

MP = magnification for the instrument

Question 29

What is the equation for the (angular) magnification for an optical instrument when placed at the eyes focal point?

Answer 29

ε = h/f = visual angle with instrument ε₀ = visual angle without instrument h = object height f = focal length s₀ = near point

Question 30

What is the equation for the numerical aperture of a magnifying glass?

Answer 30

NA = numerical aperture n = refractive index θₘₐₓ = the half-angle of the maximum light cone picked up by the objective lens.

Question 31

What is the equation for magnification by the objective lens?

Answer 31

M = magnification s’ = distance from image to lens s = distance from object to lens t = tube length f = focal length

Question 32

What is the equation for magnification by the eyepiece lens?

Answer 32

f = focal length s₀ = near point

Question 33

What is the equation for the total magnification of a microscope?

Answer 33

t = tube length f = focal length s₀ = near point

Question 34

What is the equation for the total magnification of a telescope?

Answer 34

f = focal length s₀ = near point

Question 35

What is the transfer matrix equation that changes an initial ray vector into the final ray vector?

Answer 35

u = initial vector u’ = final vector y = initial distance from the optical axis y’ = final distance from the optical axis α = initial angle with respect to the optical axis α’ = final angle with respect to the optical axis A, B, C, D = elements of the transfer matrix

Question 36

What is the equation for the transfer matrix of a complex optical system?

Answer 36

M = matrix

Question 37

What is the equation for the intensity of light?

Answer 37

I = intensity E₀ = amplitude c = speed of light ε₀ = permittivity in a vacuum

Question 38

What is the equation for the intensity of two superposed waves?

Answer 38

I = intensity E(r) = wavefunction

Question 39

What is the equation for the intensity of two superposed waves with the same frequency but different phases (E₁ = √(I₁) exp(iφ₁) and E₂ = √(I₂) exp(iφ₂))?

Answer 39

E₁ = √(I₁) exp(iφ₁) E₂ = √(I₂) exp(iφ₂) E₁* = √(I₁) exp(-iφ₁) E₂* = √(I₂) exp(-iφ₂)

Question 40

What is the interference equation for two waves with the same frequency but different phases?

Answer 40

I(r) = intensity ∆φ = phase difference

Question 41

What is the equation that relates phase difference to optical path length?

Answer 41

Phase difference = ∆φ Path difference = ∆r Wavelength = λ

Question 42

What is the path length difference for constructive interference?

Answer 42

∆r = path length difference λ = wavelength

Question 43

What is the path length difference for destructive interference?

Answer 43

∆r = path length difference λ = wavelength

Question 44

What is the path length difference for a given angle, θ, between two paths emerging from two slits?

Answer 44

∆r = path length difference d = slit separation θ = angle

Question 45

What is the equation used to find points of constructive interference on a screen?

Answer 45

y = L tanθ = position on screen (tan θ ~ sin θ ~ θ) L = perpendicular distance from slit to screen This equation uses ∆r = d sinθ

Question 46

What is the equation for phase difference in terms of wavenumber and path difference?

Answer 46

k = 2π/λ = wavenumber ∆r = path difference

Question 47

What are the Stokes relations?

Answer 47

t = transmission coefficient t’ = transmission coefficient entering from the other plane r = reflection coefficient r’ = reflection coefficient entering from the other plane

Question 48

What are the wave equations for each path in the Mach-Zehnder interferometer?

Answer 48

Question 49

Assuming that an equal amount of light is reflected and transmitted, what is the equation for the intensity on the positive and negative ports of the Mach-Zendler interferometer?

Answer 49

Question 50

How can the total intensity from the beams in a Mach-Zehnder interferometer be calculated?

Answer 50

I₀ = I₊ + I₋

Question 51

What is the condition for destructive interference with a Michelson interferometer?

Answer 51

d = distance moved = (∆L/2) ∆L = change in path length

Question 52

What is the condition for constructive interference with a Michelson interferometer

Answer 52

d = distance moved = (∆L/2) ∆L = change in path length

Question 53

How can the wavelength of light be calculated using a Michelson interferometer?

Answer 53

n = number of fringes counted d = distance moved

Question 54

What is the equation for the intensity distribution of a diffraction grating?

Answer 54

I = intensity I₀ = Amplitude k = wavenumber N = number of slits ∆r = path length difference

Question 55

What is the path length difference equation for two reflected beams from a thin film?

Answer 55

∆D = path length difference n₂ = index of refraction inside the film d = width of thin film θ₂ = angle of refraction

Question 56

What are the conditions for constructive and desctructive interference of a thin lens?

Answer 56

Top = destructive interference Bottom = constructive interference

Question 57

What is the condition for destructive interference for anti-reflection coatings?

Answer 57

∆D = path length difference

Question 58

What is the condition for constructive interference for anti-reflection coatings?

Answer 58

∆D = path length difference

Question 59

What is the equation used to find the sum of all the spherical waves arriving at a point from a single slit?

Answer 59

E₀ = amplitude b = slit width k = wavenumber r = √((x’ - x)² + z₀²)

Question 60

What is the resolution angle of a telescope?

Answer 60

R = radius of the diffracting aperture D = diameter of the diffracting aperture

Question 61

What is the equation for the numerical aperture of a microscope?

Answer 61

NA = numerical aperture D = diameter of aperture f = focal length

Question 62

What is the equation for the resolution of a microscope?

Answer 62

x = smallest resolvable distance (spatial resolution) NA = numerical aperture f = focal length D = diameter of aperture

Question 63

What is the equation for the intensity of the diffraction of a single slit?

Answer 63