Wave Optics Flashcards
(95 cards)
1
Q
Maxwell’s equations in a vacuum (q = 0, J = 0)
A
∇ × E = −∂B/∂t
∇ × B = µ0ϵ0 ∂E/∂t
∇.E = 0
∇.B = 0
2
Q
Wave equation
A
∇^2E = µ0ϵ0 ∂^2E/∂t^2
c =sqrt(1/µoϵ0)
3
Q
Plane wave equation
A
E = E0exp[i(k.r−ωt)]
4
Q
Define wavefronts
A
Wavefronts are surfaces, or loci of continuous points on the wave, with the same phase.
5
Q
Poynting vector
A
S =1/µ0 E × B.
6
Q
Define irradiance
A
Irradiance (or intensity), I, is defined as the energy in a wave crossing unit area per unit time, averaged over many 1/f intervals. It is proportional to the square of the electric field amplitude.
7
Q
k . E, k . B
A
For a wave propagating in free space, k, E, B are mutually perpendicular, and the wave is transverse.
8
Q
General solution of the 1-D wave equation
A
ψ(x, t) = f(x − vt) + g(x + vt)
the solution of the wave equation represents functions, travelling in either direction, which maintain shape as the propagate.
9
Q
Helmholtz equation for spatial part (solution to wave equation in 3-D)
A
∇^2Er = −k^2Er
A spherical wave solution (A/r) exp(i(k.r ± ωt)) is possible, more complicated solutions are possible with cylindrical symmetry.
10
Q
Huygens-Fresnel principle
A
All points on a wavefront can be considered as point sources for production of secondary spherical wavelets. At a later time the new position of the wavefront will be a surface tangent to the secondary wavelets. The wavelets have destructively interfered in all but one direction to form this new wavefront. The amplitude of the optical field at any point beyond is the superposition of all these wavelets.
11
Q
Optical path
A
If light traverses a distance l through a medium of refractive index n, the optical path is nl, or more generally integral(n(l)dl) if the refractive index is variable.
If we make light travel an extra distance l through glass of refractive index n, the extra optical path is nl. If we replace a length l of the existing path of light in air (strictly vaccuum) with length l of glass, then the extra optical path is (n-1)l.
12
Q
Fermat’s principle
A
A light ray going from A and B traverses a path that is stationary with respect to variations in path.
“light takes the minimum time to travel between A and B”, “if a particular path has a lower optical path than all other paths immediately surrounding it, light will travel along this path”
“minimum” should be read as “minimum or maximum or saddle point”, a stationary point, or locally non-varying region of optical path.
13
Q
Geometrical Optics
A
Understand by drawing wavefronts and rays, or by Fermat’s principle.
14
Q
Law of reflection
A
θi = θr
15
Q
Law of refraction (Snell’s law)
A
n1 sin(θi) = n2 sin(θr)
16
Q
Critical Angle
A
If light travelling in an optically dense medium is incident on a boundary with a less dense medium, there will be some incidence angle θc for which the diffraction angle is θr = 90◦. This angle θc is known as the critical angle. Light incident at θ > θc will not pass into the less dense medium, and instead will be totally internally reflected back into the denser medium.
17
Q
Paraxial approximation
A
Often used in geometric optics, the approximation states that all process take place close to the optic axis of the system. Consequently, all angles are small so that θ ≃ sin θ ≃ tan θ.
18
Q
Reflection at a spherical surface
A
1/u + 1/v = 2/R
19
Q
Refraction at a spherical surface
A
ni/u + nr/v = (nr − ni)/R
20
Q
Snellian focusing of light
A
The lens is shaped so that rays emerging from the object O will be presented with surfaces at which they refract. If we shape the lens correctly, we can arrange for all such refracted rays to end up at the same end point, I.
21
Q
Fermatian focusing of light
A
We arrange the shape of the lens such that the rays going from the object O to the image I will all traverse the same optical path. Thus, if we converge the rays with a convex lens, the rays along the optic axis will traverse a thicker part of the glass, and this extra optical path exactly compensated for the extra geometrical path encountered by rays further from the axis.
22
Q
Lens maker equation
A
1/f = (n-1)(1/R1 - 1/R2)
(u1 → ∞ as incoming rays are parallel, then v2 is the focal length, f)
22
Q
Lens formula
A
1/u1 + 1/v2 = (n-1)(1/R1 - 1/R2)
23
Q
Lens equation
A
1/f = 1/u + 1/v
24
Focal length
The focal length of a lens, f, is the distance from a lens at which an image is formed, if the object is at infinity.
25
u and v convention
the object distance u is positive if the object is to the left (in front of) the lens, and the image distance v is positive if the image is to the right of (behind) the lens.
26
Converging and diverging lens
A converging lens, with positive f, causes incident parallel rays to converge.
A diverging lens, with negative f, causes incident parallel rays to diverge.
27
Radii of curvature convention
Radii of curvature are positive if convex, as seen from the front (incoming ray). Known as the 'Gaussian' convention.
28
Real or virtual image
An image is real if it can be formed on a screen (rays intersect at the image).
It is virtual is it cannot (rays will intersect at the image only if extrapolated from other rays).
29
How to draw ray diagrams
* Draw the lens and optic axis: draw an object in front of the lens.
* Draw a ray from the object through the centre of the lens: this ray is not deviated.
* Draw a ray from the object parallel to the optic axis. In a converging lens, this is refracted
through the focal point on the far side. In a diverging lens, this is refracted away from the
optic axis along a line that, projected back, originates at the focal point on the near side.
* If the lines you have drawn cross, the image is real and at the position and height implied by
your lines.
* If they do not cross, extrapolate one or both backwards to find the position and size of the
resulting virtual image.
30
Magnification of images
The magnification of an optical system is the ratio of the size of an image to the size of the object creating it.
31
Gravitation lensing and optics (Snellian)
According to general relativity, mass bends light. Light from a background source is deflected by an intervening mass and rays arrive at the observer along different light paths, resulting in multiple imaging. We can calculate the positions of the images by working out the binding angles.
32
Gravitational lensing and optics (Fermatian)
Light going through a gravitational potential suffers a delay ("Shapiro delay"). First order understanding: light tries to find a compromise minimum path between the direct path (short geometric but lots of Shapiro delay) and a very bent path (long geometric path but less Shapiro delay). Path chose is the minimum in between these two.
33
Convention for circular polarisation
RCP is defined as the electric vector rotating clockwise when viewed from the observer i.e. looking backwards along the wave.
33
Stokes parameters
I
Q
U
V
34
Electric field for polarization
E = Ex xˆ + Ey yˆ
= xˆE0x cos(kz − ωt) + yˆE0y cos(kz − ωt − ϵ).
ϵ is the dierence between the two phase angles ϕ2, ϕ1.
35
How is polarized light produced by reflection?
* Incident light's E-vector excited oscillations of dipoles in dielectrics along the direction of the E-vector (not direction of propagation)
* These dipoles do not radiate along their length but do in other directions.
* Can resolve amplitudes of E-vectors along different directions as required; recall that intensity is proportional to the square of the amplitude.
36
Brewster angle
A totally polarised reflected ray is produced when tan(θi) = nr/ni. This value of θi = θB, is the Brewster angle.
37
What are the fresnel coefficients
* We want to derive the strengths of the two polarization components in the reflected ray, not just for the Brewster angle.
* The coefficients of the reflected and transmitted (refracted ray), in terms of the incident amplitude, are the Fresnel coefficents.
* Fresnel coefficients usually derived directly from Maxwell's equations with EM arguments about the boundary conditions.
38
Jargon for polarized light (parallel and perpendicular component)
The component polarized perpendicular to the plane (i.e. a and b) is known as the s- polarized light, and the parallel component (A and B) is known as the p- polarized light.
39
Amplitude Fresnel coefficients (reflected/incident) for s and p polarized light
fs = −sin(θi − θr)/sin(θi + θr)
fp = tan(θi − θr)/tan(θi + θr)
40
Production of polarized light by scattering
* incoming light has E-vector oscillation perpendicular to the direction of travel - resolvable into two orthogonal directions,
* these excite dipoles to oscillate in these directions
* the resulting re-radiated light is not emitted along the direction of oscillation.
41
Malus' law (intensity transmitted in polarization)
The intensity transmitted by a dichroic when plane-polarized light is incident on it is I(θ) = I0 cos^2(θ), where θ is the angle between the initial E-vector and the pass axis.
41
Transmission of dichroic material
Counter-intuitive, analogy of prison bars is not a good analogy.
An oscillation parallel to the long axis of the molecules will be absorbed, by exciting electrons moving along the long axis.
42
Anisotropic crystal
An anisotropic crystal (also known as a birefringent crystal) is a crystal in which the refractive index n depends on the direction of oscillation of the E-vector of the wave.
43
Corollary for optical anisotropy
The velocity of the wave, and the relative permittivity ϵ also depend on the direction of oscillation of the E-vector.
It is easier to polarize an anisotropic crystal in one direction relative to another.
44
Uniaxial crystal
A uniaxial crystal is one for which two of the components of ϵ are equal and the other differs. Because ϵ ∝ n^2 this means that the refractive index (and hence velocity of propagation of light with that polarization ) differs in one direction from the other.
45
Biaxial Crystal
A biaxial crystal is one for which all three components of ϵ are different.
46
Negative uniaxial crystal
A negative uniaxial crystal is one for which the unique axis has a lower refractive index (and hence higher velocity, i.e. a "fast axis".
47
Solutions to the wave equation for a birefringent crystal
a very standard one:
k^2/no^2 = w^2/c^2
and a strange solution:
kx^2/ne^2 + ky^2/ne^2 + kz^2/no^2 = w^2/c^2
here the z axis is special, no is the ordinary refractive index, ne is the extraordinary refractive index.
48
Refractive index of birefringent material
1/n^2 = sin^2(θ)/ne^2 + cos^2(θ)/no^2
49
Structure of anisotropic crystals
For a uniaxial crystal the different refractive index applies to light with the E-VECTOR in the special directions (optic axis), NOT the direction of propagation.
For light propagating along the optic axis, BOTH components of polarization will have E-vectors along the "normal" directions and no special effects will be apparent.
50
Anisotropic crystals vs dichroics
Dichroics influence polarization state by selective absorption of one polarization (and thus affect density). Anisotropic crystals do not absorb, but provide different refractive indices for different polarization states.
51
Ordinary and extraordinary rays
The wave in a uniaxial crystal moving at the velocity common to waves with E-vector along two axes is know as the ordinary ray.
The wave moving at the velocity corresponding to the other axis is known as the extraordinary ray.
For a negative uniaxial calcite, the extraordinary ray velocity is faster, and the corresponding refractive index ne is therefore lower than no.
52
Half Wave Plate HWP
Do not confuse the relative phase shift between o and e-rays produced by a half wave plate, with the angle of rotation of linearly polarized light produced by HWP.
53
How to produce a quatre wave plate QWP
There needs to be a relative shift in optical path between o and e-waves of λ/4. This shift in optical path is also given by |dne−dno|. So the thinnest slab of birefringent material which can be used as a QWP is
d = λ/(4|ne − no|).
For the HWP, a factor of 2 instead of 4.
54
Glan polarizers vs Wollaston prisms
Look similar but mode of operation is different
55
Quartz wedge
Optic axis along side of wedge, relative delay of light incident on the wedge at different points.
If we put an analysing polarizer above the wedge, with a pass direction at −45◦, we get a dark band
at D and the origin of the wedge, and a bright band at B. λ/4 circular polarisation, λ/2 bright band , 3λ/4 circular polarisation and λ dark band.
56
Faraday effect
The rotation of the plane polarization of light passing through a magnetic field.
θ = VBd.
Direction of rotation depends on the sign of V, think of the wrapping of a current around a B-field.
If a wave propagates through a B-field and comes back the same way, the rotation doesn't cancel, it doubles.
57
Define interference
Interference refers to the combination of finite numbers of waves (such as the two combining waves from the Young's slits experiment; diffraction refers to the combination of an infinite number (such as those from different parts of a wide aperture). We treat them separate, but the principles are the same. In particular, the resultant field is the sum of the combining fields, and the intensity is proportional to Etot*.Etot.
58
Intensity and interference intensity equation
E∗.E = |E1|^2 + |E2|^2 + 2E1.E2 cos[(k1 − k2).r − (ω1 − ω2)t + (ϵ1 − ϵ2)]
Not just sum of individual intensities, third term is the interference term.
59
Coherence conditions
*⟨E1.E2⟩ ̸= 0. Equivalent to demanding that interference must not be orthogonal. E.g. if two waves linearly polarized at 90◦, they are orthogonal and interference term is zero.
* ω1 ≃ ω2. Needed because we observe the intensity pattern averaged over a time interval. Unless angular frequencies are equal a term involving cos(ω1 − ω2)t will quickly average to zero. We need ∆ω < 1/T where T is the time over which we observe.
* Averaged over time, we require ϵ1 − ϵ2 is constant, i.e. waves have a constant phase relation over time.
We add waves by adding amplitudes and forming E*.E. Otherwise (no coherence, interference terms average to zero), we can add intensities.
60
Phasors
As with vectors, the angle of each phasor is the angle to the real (x) axis, not the previous phasor.
61
Temporal coherence
Temporall coherence is the measure of the average correlation of the phase of the light wave along the propagation direction. The temporal coherence time is given by tc = 1/∆ν, and if t > tc interference will not be observed between a wave and its delayed version. The corresponding coherence length is ctc.
Temporal coherence is important for division of amplitude interferometry.
62
Two waves with slightly different frequencies get out of phase in time ...
tc = 1/∆ν
63
Spatial coherence
Spatial coherence is the correlation of the phase of a light wave transverse to the propagation direction (how uniform is the wavefront).
64
Combined wave equation of Young's slits and assumptions
Write the combined wave as
Etot = E exp[i(kx−ωt)] + E exp[i(kx−ωt+kd sin θ)]
* Assumed that ω1 = ω2, k1 = k2 and ϵ1 −ϵ2 =
kd sin θ, and that the amplitudes from each
slit are equal.
* Assumed that distance to screen is large, so
interfering waves are parallel, and the distances from the slits to the screen are equal apart from the
distance d sin θ. So k.(r1 − r2) = kd sin θ.
65
Interference condition (Young's slits)
A set of bright fringes occur when kdy/2s = nπ, or when y = nλs/d, or when d sin θ = nλ.
66
Visibility of a set of fringes
V = Imax - Imin /(Imax + Imin)
Imax and I min are the maximum and minimum of the intensity pattern. In the Young's slits case V = 1 because Imin = 0, not always the case.
67
Lloyds Mirror
There is a phase change of π on reflection from an optically denser surface.
68
N slits intensity pattern
I = E0^2 (sin^2(Nkδ/2)/sin^2(kδ/2))
= E0^2 (sin^2(Nπyd/λ)/sin^2(πyd/λ))
69
Michelson interferometer path difference
∆ = 2(d1 − d2), d1 and d2 are mirror movements.
70
Thin film interference optical path difference and interference condition
path difference = 2dn cos(θr)
θr is the angle of refraction,d is the width of the film, n is the refractive index of the film.
For normal incidence the optical path difference is 2dn.
Interference condition is
2dn cos θr = (m − 1/2)λ
71
Fabry-Perot etalon (phase shift)
Each wave has a phase shift δ = (2π/λ).2d cos(θ) with respect to the last. The derivation is exactly the same as for thin films.
No n in the phase shift, paths involved in the path difference are both in air. Interference condition is
δ = 2mπ for a maximum, since there are two extra reflections from the denser media, and hence two π phase shifts, between each successive wave.
72
Free spectral range of an etalon
The free spectral range of an etalon is the spacing between individual principal interference peaks,
∆ν, ∆λ.
73
Resolution of an etalon
The resolution of an etalon is the width of each principal interference peak, δν, δλ.
74
Finesse of an etalon
The finesse of an etalon is the ration of free spectral range to resolution, it represents how many individual spectral lines can be resolved without confusion.
75
Diffraction strategy
* Same principals as interference, except considers interference of large (infinite) number of parts of waves rather than small/finite number.
* Derived from the wave equation/Helmholts equation.
* Start assuming far-field conditions - need Fourier transforms.
* Restrictive approximation on HF integral to study near-field (Fresnel) diffraction.
76
Fourier Series
77
Fourier transforms
f(x) = 1/2π integral(-∞ to ∞)F(ξ)exp[iξx] dξ
F(ξ) = integral(-∞ to ∞)f(x)exp[−iξx]dx.
78
Principles of Fourier transforms
* Small and spikey goes to broad and smooth.
* Widen a function, its FT shrinks.
* FT of a function tells you about the range of frequencies present. Something small and spikey has lots of high frequencies.
79
Fraunhofer (far-field) diffraction (difference from interference)
Addition of waves the same as for interference, instead of adding a few waves of finite amplitude, we add very many small waves with different phases, sums become integrals.
80
Assumptions of Fraunhofer (far-field) diffraction
* Phase varies linearly across the aperture - assume screen is large distance away.
* The amplitude of arriving waves at the screen is the same for all parts of the aperture - true if screen is much more distant than size of aperture.
* Ignored obliquity factor (from HF integral).
* Source is a long way from the aperture, no amplitude or phase variation across the slit.
λ ≪ R
s, r constant in denominator
K(α) constant
x, y ≪ X, Y
phases linear across aperture
s = s0 in phase term (plane wave)
81
General principle of phase shift
If part of the aperture has a ϕ phase
shift, treat it separately and multiply the integrand by
exp(iϕ).
82
Define convolution
Convolution of two functions f(x) and g(x) is defined as
h(x) = integral(-∞ to ∞)f(x')g(x − x')dx'.
83
Convolution theorem
If the function h(x) is the convolution of two functions f(x) and g(x), then the Fourier transform h(x) is the product of the Fourier transforms of f(x) and g(x).
84
Convolution of apertures
The diffraction pattern is the Fourier transform of the complex aperture distribution. Hence, if an aperture can be made up as the convolution of two other apertures, then the diffraction pattern is the product of the individual diffraction patterns.
85
Comparison of the FP etalon and diffraction grating
Diffraction grating : Fabry-Perot etalon
Resolving power λ/δλ mN ∼ 10^5 : πm√ρ/(1 − ρ) ∼ 10^6
Free spectral range ∆λ λ/m ∼ 100nm : λ/m ∼ 0.01nm
Finesse ∆λ/δλ N ∼ 10^4 : π√ρ/(1 − ρ) ∼ 100
86
Intensity diffraction pattern from a circular aperture
A uniformly illuminated circular aperture produces a circularly-symmetric Airy function intensity diffraction pattern. The function has a radius to the first dark fringe of 1.22 λ/d and a width of
∼ λ/d
where d is the diameter of the circular aperture, and this width is in radians. You can use λ/d when asked for the resolution.
87
Huygens-Fresnel integral
E(P) = ϵ′exp[iks]/s integral (K(α) exp[ikr] / r )dA
88
Rayleigh distance
The distance at which non-linear effects become important. It is usually defined as
2y^2/λ,
where y is the size of the aperture. Far-field diffraction theory is valid if we are at a much greater distance than this from the aperture.
89
Fresnel (spherical on axis) assumptions
λ ≪ R, R is the distance from aperture to screen.
90
Fresnel (straight-edge on-axis) assumptions
λ ≪ R
s, r constant in denominator -
K(α) constant - obliquity factor
91
Stimulated emission cross section
σ ≡ sB21hν/c
92
Gain coefficient
γ is the gain coefficient. The system has optical gain if
γ > 0, which happens if N2 > N1 (population inversion).
