Optics: Equations Flashcards
What is Maxwell’s equation for the (charged) source of an electric field in a vacuum?
∇ = del operator
E = electric field
What is Maxwell’s equation for the (charged) source of a magnetic field in a vacuum?
∇ = del operator
B = magnetic field
What is Maxwell’s equation for electromagnetic induction (Faraday’s Law)?
∇ = del operator
E = electric field
B = magnetic field
What is Maxwell’s equation for Ampere’s Law?
∇ = del operator
B = magnetic field
E = electric field
ε₀ = dielectric constant in a vacuum
µ₀ = permeability in a vacuum
c = speed of light
What is the wave equation for an electric field?
∇ = del operator
E = electric field
c = speed of light
What is the wave equation for a magnetic field?
∇ = del operator
B = magnetic field
c = speed of light
What is the equation for the wavelength in a vacuum?
λ₀ = wavelength in a vacuum
c₀ = speed of light in a vacuum
ν = frequency
What is the equation for the wavelength in a medium?
λ’ = wavelength in a medium
c’ = speed of light in a medium
ν = frequency
What is the solution to the wave equation in one dimension?
E₀ = amplitude (could also be B₀ for a magnetic wave)
k = wavenumber = 2π/λ
ω = angular frequency = 2πν
What is the solution to the wave equation in three dimensions?
E₀ = amplitude
k = wave vector
ω = angular frequency = 2πν
What is the solution of the wave equation for a plane wave?
E₀ = amplitude
r = (x, y, z)
k = wave vector
ω = angular frequency = 2πν
c.c = complex conjugate
What is the solution of the wave equation for a spherical wave?
E₀ = amplitude
r = |r| = √(x² + y² + z²)
k = wave vector
ω = angular frequency = 2πν
c.c = complex conjugate
What is the equation for the Poynting vector?
S = Poynting vector
E = electric field
B = magnetic field
µ₀ = permeability in a vacuum
What is the equation for the index of refraction?
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
What is the equation for Snell’s Law?
n₁ = refractive index of medium 1
n₂ = refractive index of medium 2
θ₁ = angle of incidence
θ₂ = angle of refraction
What is the equation for the critical angle?
What is the equation for Fermat’s Principle?
t = time
d = distance
nd = L = optical path length (n = local refractive index)
c’ = speed in a medium
c = speed in a vacuum
What is the equation used to calculate the optical path length?
The minimal value of this equation is found.
L(r) = optical path length
n(r) = refractive index
A = initial point
B = final point
What is the imaging equation?
s = distance from object to optical element
s’ = distance from image to optical element
f = r/2 = focal length
What is the equation for magnification?
M = magnification
h’ = image height
h = object height
s’ = distance from image to optical element
s = distance from object to optical element
What is the imaging equation for image formation at a spherical surface?
n₁, n₂ = refractive index
s = distance from object to optical element
s’ = distance from image to optical element
What is the magnification equation for a spherical refractive surface?
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
What is the focal length of a thin lens that is curved on both sides?
f = total focal length
f₁ = focal length (side 1)
f₂ = focal length (side 2)
What is the imaging equation for a thin lens in a medium?
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
What is Lensmaker’s equation?
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
What is the magnification equation for a thin lens?
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
What is the equation for the visual angle (or angular diameter / apparent size) of an object?
h = object height
ε₀ = visual angle
What is the equation for the (angular) magnification for an optical instrument?
MP = magnification for the instrument
What is the equation for the (angular) magnification for an optical instrument when placed at the eyes focal point?
ε = h/f = visual angle with instrument
ε₀ = visual angle without instrument
h = object height
f = focal length
s₀ = near point
What is the equation for the numerical aperture of a magnifying glass?
NA = numerical aperture
n = refractive index
θₘₐₓ = the half-angle of the maximum light cone picked up by the objective lens.
What is the equation for magnification by the objective lens?
M = magnification
s’ = distance from image to lens
s = distance from object to lens
t = tube length
f = focal length
What is the equation for magnification by the eyepiece lens?
f = focal length
s₀ = near point
What is the equation for the total magnification of a microscope?
t = tube length
f = focal length
s₀ = near point
What is the equation for the total magnification of a telescope?
f = focal length
s₀ = near point
What is the transfer matrix equation that changes an initial ray vector into the final ray vector?
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
What is the equation for the transfer matrix of a complex optical system?
M = matrix
What is the equation for the intensity of light?
I = intensity
E₀ = amplitude
c = speed of light
ε₀ = permittivity in a vacuum
What is the equation for the intensity of two superposed waves?
I = intensity
E(r) = wavefunction
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))?
E₁ = √(I₁) exp(iφ₁)
E₂ = √(I₂) exp(iφ₂)
E₁* = √(I₁) exp(-iφ₁)
E₂* = √(I₂) exp(-iφ₂)
What is the interference equation for two waves with the same frequency but different phases?
I(r) = intensity
∆φ = phase difference
What is the equation that relates phase difference to optical path length?
Phase difference = ∆φ
Path difference = ∆r
Wavelength = λ
What is the path length difference for constructive interference?
∆r = path length difference
λ = wavelength
What is the path length difference for destructive interference?
∆r = path length difference
λ = wavelength
What is the path length difference for a given angle, θ, between two paths emerging from two slits?
∆r = path length difference
d = slit separation
θ = angle
What is the equation used to find points of constructive interference on a screen?
y = L tanθ = position on screen (tan θ ~ sin θ ~ θ)
L = perpendicular distance from slit to screen
This equation uses ∆r = d sinθ
What is the equation for phase difference in terms of wavenumber and path difference?
k = 2π/λ = wavenumber
∆r = path difference
What are the Stokes relations?
t = transmission coefficient
t’ = transmission coefficient entering from the other plane
r = reflection coefficient
r’ = reflection coefficient entering from the other plane
What are the wave equations for each path in the Mach-Zehnder interferometer?
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?
How can the total intensity from the beams in a Mach-Zehnder interferometer be calculated?
I₀ = I₊ + I₋
What is the condition for destructive interference with a Michelson interferometer?
d = distance moved = (∆L/2)
∆L = change in path length
What is the condition for constructive interference with a Michelson interferometer
d = distance moved = (∆L/2)
∆L = change in path length
How can the wavelength of light be calculated using a Michelson interferometer?
n = number of fringes counted
d = distance moved
What is the equation for the intensity distribution of a diffraction grating?
I = intensity
I₀ = Amplitude
k = wavenumber
N = number of slits
∆r = path length difference
What is the path length difference equation for two reflected beams from a thin film?
∆D = path length difference
n₂ = index of refraction inside the film
d = width of thin film
θ₂ = angle of refraction
What are the conditions for constructive and desctructive interference of a thin lens?
Top = destructive interference
Bottom = constructive interference
What is the condition for destructive interference for anti-reflection coatings?
∆D = path length difference
What is the condition for constructive interference for anti-reflection coatings?
∆D = path length difference
What is the equation used to find the sum of all the spherical waves arriving at a point from a single slit?
E₀ = amplitude
b = slit width
k = wavenumber
r = √((x’ - x)² + z₀²)
What is the resolution angle of a telescope?
R = radius of the diffracting aperture
D = diameter of the diffracting aperture
What is the equation for the numerical aperture of a microscope?
NA = numerical aperture
D = diameter of aperture
f = focal length
What is the equation for the resolution of a microscope?
x = smallest resolvable distance (spatial resolution)
NA = numerical aperture
f = focal length
D = diameter of aperture
What is the equation for the intensity of the diffraction of a single slit?
