General properties of light are its ________ and its ______ such that the angle of ______ (as measured with respect to the surface normal) is _______________________________.
- rectilinear propagation
- equal in magnitude to its angle of incidence.
Refraction of light is governed by _______, ____________, with θi,t and ni,t, respectively, the angle of incidence (i) / transmission (t) and the index of refraction of the medium of incidence (i) / transmission (t).
- Snell’s law
- ni sin θi = nt sin θt
• The propagation of light follows _________, which states that _____________________________________________.
- Fermat’s theorem
- the actual path between two points taken by a beam of light is the one which is transversed in the least time
It can also be described according to ___________, in that ________________________________________________________.
- Huygens’s principle
- every point on a wavefront acts as a source of a new wavefront propagating radially outwards
Optical path length is defined as___________________________________________. As a consequence, the speed of light in such a medium is given by ___.
- the distance that light travels in a vacuum in the same time that it travels a distance d in a medium with index of refraction n
Total internal reflection occurs when _____________________________________ . This is the case for transmission from an optically denser (larger n) to an optical less dense (smaller n) medium for angles of incidence larger than the critical angle θc =_______.
- the angle of incidence is so large that the angle of transmission, by virtue of Snell’s law, would become larger than 90o
- sin−1 (nt/ni)
Light paths are reversible. That is, _________________________________________________________.
- the path of light from A to B will be identical to the (reverse) path from B to A
Light can be described as a wave ψ = Ae i(k·r−ωt) . By ignoring polarisation and considering only one-dimensional propagation along the positive x direction, this can be simplified to ψ = ________.
In general, the intensity of light for a given polarisation direction is proportional to _________, where the asterisk * refers to the __________.
- |ψ|2 = ψ ∗ψ
- complex conjugate
For light propagating in a medium of index of refraction n, we need to substitute c → c/n, hence λ = c/f → (c/n)/f = λ/n and k = 2π/λ → kn. As a consequence, the position-dependent part of the phase becomes knx, which is equivalent to the usual expression for the wave with x → nx, i.e., what matters is the optical path nx.
In Young’s double slit experiment, _______________________________________________, resulting in an______________ on a detection screen at a distance L. For any point on the screen, the light can interfere _______________ or ____________, depending on the _____________________ to that point.
- monochromatic light passes through two narrow slits with separation d
- interference pattern
- destructively or constructively
- difference in optical path from each of the slits
Two light rays will interfere destructively if their path difference is_______. This corresponds to a phase difference at a particular point of _______, with m integer. Similarly, they will interfere constructively if their path difference is __, i.e., their phase difference at a particular point is m 2π.
- (m+ 1/2 )λ
- (m+ 1/2 )2π
In the Fraunhofer limit, _____________________________________________, in which case the relevant optical path difference is given by_____, where θ is the angle of detection.
- the light rays from the two slits are presumed to propagate (approximately) parallel to a point on the screen
- d sin θ
We can write ψ1 = E e(ikr1 − iωt) and ψ2 = E e(ikr1 + iδ − iωt) for the light waves originating from the two slits, where δ = ______. We can calculate the light intensity at a particular point I ∝ ψ∗ψ by noting that the total ψ = A e−iωt = (E eikr1 + E eikr1+iδ) e−iωt, hence I ∝ A∗
- kd sin θ
Generally, we determine an interference (or diffraction) pattern by __________________________, to yield the resulting complex __________. In the example of the Young’s double slit experiment, we find A = _________________, with δ = ______, and I ∝ ____________________.
- summing waves in complex exponential notation
- amplitude A
- A = 2E eikr1+iδ/2 cos (δ/2)
- kd sin θ
- 4E2 cos2(πd sinθ/ λ)
The Young’s double slit experiment is an example of ___________, and can be expanded by, e.g., considering the case where the amplitudes of light through slit 1 and slit 2 are not equal, to yield a lower visibility of the fringes V = __________________
- interference by wavefront division
- (Imax − Imin)/(Imax + Imin)
An alternative form of interference is by ________, e.g., ________________
- amplitude division
- double reflection from a thin film
Generally speaking, constructive interference results when the (optical) path difference for light following two different path Γ = __, with integer m; and destructive interference when Γ = ______.
- Γ = mλ
- Γ = (m + 1/2)
Upon reflection from the boundary towards an optically denser medium (__________), the light acquires a phase shift of __, equivalent to an additional ____; hence in general, phase shift on reflection may be relevant for determining conditions of constructive and destructive interference.
- from lower to larger n
- 1/2 λ
For a thin film, one can show that Γ = _______, with d ___________, n its __________, and β_____________ (with respect to the surface normal) in the film. Hence for perpendicular incidence (cos β = 1) Γ is simply n × 2× the film thickness.
- 2nd cos β
- the thickness of the film
- its index of refraction
- the angle of the light path
Relevant examples of thin film interference are ____________________________________.
anti-reflection coatings, wedges, and Newton’s rings.
Spatial coherence is __________________________________________, i.e., over what distance the phase change is given by ∆φ = (2π/λ)∆x
- the distance one goes along a wave with constant separation of peaks and troughs
Temporal coherence is _____________________________________________________________________________________________________, i.e., for how long ∆φ = ω∆t.
a measure of how long a wave at a particular point maintains a distinct phase relationship between equivalent parts of the wave
In the Michelson interferometer, an appropriate arrangement of mirrors allows light to follow two different paths that are next recombined, and interference depends on the ______________, as usual, and the interference pattern can be calculated by __________________________________.
- optical path difference
- summing the waves in complex notation.
In a Fabry-Perot interferometer, we have a special case of thin film interference, but with (generally) a reflection coefficient that is so high that multiple reflections between the two surfaces contribute significantly to the resulting interference pattern. The higher the number of sizeable reflections, the sharper the interference pattern becomes.
- sharper the interference pattern becomes
One can define the resolving power of an interferometer as λ/∆λ, for separating two wavelength λ and λ + ∆λ. For the Fabry-Perot interferometer, this can be calculated by _______________________________________________________________.
- considering when the peak of one interference pattern (e.g., for λ + ∆λ) coincides with the half-maximum of the other (e.g., for λ).
For diffraction of light, one needs to calculate a (generally complex) _________, resulting from the _________originating from different parts of an aperture or from a grating.
- total amplitude A
- sum of waves
For a grating of N narrow slits in the Fraunhofer limit, A = ___________________ , with δ = ______. Note that the Young’s double slit experiment is a special case of this, with N = 2.
- Eeikr1 (1 + e iδ + (e iδ)2 + . . .(e iδ) N−1 )
- kd sin θ
This leads to an intensity I(θ) =___________, where x = (π/λ)d sin θ, where one can determine primary maxima, zeros, and subsidiary maxima. As for the Fabry-Perot interferometer, the ______the number of waves N that contribute to the pattern (as a particular point of detection), the sharper the diffraction pattern.
- I0sin2 Nx/sin2 x
The diffraction pattern for a slit of finite width a follows by considering that slit as a diffraction grating with N narrow slits separated by d = a/N, and next letting N → ∞. The intensity of the diffraction can thus be deduced from the grating result, yielding I = _______ , with now x = ______
- I0sin2 x/x2
- (π/λ)a sin θ
The Rayleigh criterion states that two light sources diffracted through, e.g., a slit of width a can just be resolved behind the slit if their diffraction patterns are separated at least so far that the ______________________________________________________. This yields a minimum resolvable (Rayleigh) angle θR = ___. For a disk-shaped aperture of diameter D, θR ________.
- (primary) maximum of one coincides with the first zero of the other
- = 1.22 λ/D
Light is a form of electromagnetic radiation, with 400
- electric field
- magnetic field
The direction of E determines the _________ of the light, which can be ______ (E oscillates in a single, well-defined direction), ______ (E rotates as a function of position and time, while retaining its magnitude), and ______ (as circular, but now the magnitude of E is not the same in all directions).
- type of polarisation
Circularly and elliptically polarised light can be described as _________________________. If a wave is approaching an observer and E appears to rotate clockwise or counterclockwise for the observer, we speak, respectively, of right-handed and left-handed circular (or elliptical) polarisation.
- two orthogonal components of E that are 90o out of phase
In general, the reflection and transmission coefficients for light at a boundary (Fresnel’s equations) depend on the polarisation. When the angle of incidence equals the Brewster angle ________, no light is reflected with polarisation parallel to the plane of incidence, so the reflected light is linearly polarised in the direction perpendicular to this plane
- θp = tan−1 (nt/ni)
Light can be polarised by ____________, by making use of materials than have an adsorption coefficient or index of refraction that depends on the polarisation direction referred to intrinsic crystalline axes in the material, or by scattering. Some forms of light are inherently polarised by the way they are created (e.g., laser light, computer screens)
- reflection from a surface
The Law of Malus states that the transmitted light through an (ideal) polariser is given by I = _______, where I0 is the _____________ and θ the _____________ with respect to the polarisation axis of the polariser.
- I0 cos2 θ
- intensity of the incident light
- angle of the polarisation
The lens (and mirror) equation reads_______ , where p is the _________, q is the ________, and f the _____________. It is valid for thin lenses and in general for small angles. The magnification of a lens (and mirror) follows from M = −q/p.
- 1/p + 1/q = 1/f
- object distance
- image distance
- focal distance of the lens (or mirror)
For systems of lenses, one can apply the lens equation subsequently to the respective lenses through which the light passes, with the image of the first lens being the object for the second, the image of the second lens the object for the third, etc.
Sign conventions are that (i) p > 0 if the object is in front of the lens/mirror and p < 0 if it is behind it; (ii) q > 0 if the image is behind the lens / in front of the mirror and q < 0 if it is in front of the lens / behind the mirror; (iii) f > 0 for a converging lens/mirror, and f < 0 for a diverging lens/mirror.
With these sign conventions q < 0 implies that the different light paths from a point of the object do not converge to a single point in the image (as for the focussing of a real image), but the extrapolations of these light paths do, to form a ___________.
- virtual image
• These concepts of geometrical optics can be used to describe the eye and the effect of glasses, and to describe microscopes and telescopes. _________poses a fundamental limit of the resolution of optical instruments via the Rayleigh criterion, with D the optical aperture of the instrument.
Approximating the hypergeometric with the Binomial:
when the ratio n/N is small, the hypergeometric (n, M, N) distribution may be approximated by the binomial (n, p) distribution with p = M/N.