Light and Optics

Question 1

General properties of light are its ________ and its ______ such that the angle of ______ (as measured with respect to the surface normal) is _______________________________.

Answer 1

• rectilinear propagation
• reflection
• reflection
• equal in magnitude to its angle of incidence.

Question 2

Refraction of light is governed by _______, ____________, with θ_i,_t and n_i,_t, respectively, the angle of incidence (i) / transmission (t) and the index of refraction of the medium of incidence (i) / transmission (t).

Answer 2

• Snell’s law
• n_i sin θ_i = n_t sin θ_t

Question 3

• The propagation of light follows _________, which states that _____________________________________________.

Answer 3

• 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

Question 4

It can also be described according to ___________, in that ________________________________________________________.

Answer 4

• Huygens’s principle
• every point on a wavefront acts as a source of a new wavefront propagating radially outwards

Question 5

Optical path length is defined as___________________________________________. As a consequence, the speed of light in such a medium is given by ___.

Answer 5

• 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
• c/n

Question 6

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 θ_c =_______.

Answer 6

• the angle of incidence is so large that the angle of transmission, by virtue of Snell’s law, would become larger than 90^o
• sin⁻¹ (n_t/n_i)

Question 7

Light paths are reversible. That is, _________________________________________________________.

Answer 7

• the path of light from A to B will be identical to the (reverse) path from B to A

Question 8

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 = ________.

Answer 8

• Ae^i(kx−ωt)

Question 9

In general, the intensity of light for a given polarisation direction is proportional to _________, where the asterisk * refers to the __________.

Answer 9

• |ψ|² = ψ ∗ψ
• complex conjugate

Question 10

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.

Question 11

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.

Answer 11

• monochromatic light passes through two narrow slits with separation d
• interference pattern
• destructively or constructively
• difference in optical path from each of the slits

Question 12

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.

Answer 12

• (m+ 1/2 )λ
• (m+ 1/2 )2π
• mλ

Question 13

In the Fraunhofer limit, _____________________________________________, in which case the relevant optical path difference is given by_____, where θ is the angle of detection.

Answer 13

• the light rays from the two slits are presumed to propagate (approximately) parallel to a point on the screen
• d sin θ

Question 14

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 e^ikr₁ + E e^ikr1+iδ) e^−iωt, hence I ∝ A^∗

Answer 14

• kd sin θ

Question 15

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 ∝ ____________________.

Answer 15

• summing waves in complex exponential notation
• amplitude A
• A = 2E e^ikr1+iδ/2 cos (δ/2)
• kd sin θ
• 4E² cos²⁽πd sinθ/ λ)

Question 16

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 = __________________

Answer 16

• interference by wavefront division
• (I_max − I_min)/(I_max + I_min)

Question 17

An alternative form of interference is by ________, e.g., ________________

Answer 17

• amplitude division
• double reflection from a thin film

Question 18

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 = ______.

Answer 18

• Γ = mλ
• Γ = (m + 1/2)

Question 19

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.

Answer 19

• from lower to larger n
• π
• 1/2 λ

Question 20

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.

Answer 20

• 2nd cos β
• the thickness of the film
• its index of refraction
• the angle of the light path

Question 21

Relevant examples of thin film interference are ____________________________________.

Answer 21

anti-reflection coatings, wedges, and Newton’s rings.

Question 22

Spatial coherence is __________________________________________, i.e., over what distance the phase change is given by ∆φ = (2π/λ)∆x

Answer 22

• the distance one goes along a wave with constant separation of peaks and troughs

Question 23

Temporal coherence is _____________________________________________________________________________________________________, i.e., for how long ∆φ = ω∆t.

Answer 23

a measure of how long a wave at a particular point maintains a distinct phase relationship between equivalent parts of the wave

Question 24

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 __________________________________.

Answer 24

• optical path difference
• summing the waves in complex notation.

Question 25

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.

Answer 25

• sharper the interference pattern becomes

Question 26

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 _______________________________________________________________.

Answer 26

• λ/∆λ
• 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).

Question 27

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.

Answer 27

• total amplitude A
• sum of waves

Question 28

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.

Answer 28

• Ee^ikr₁ (1 + e ^iδ + (e ^iδ)² + . . .(e iδ) ^N−1 )
• kd sin θ

Question 29

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.

Answer 29

• I₀sin² Nx/sin² x
• higher

Question 30

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 = ______

Answer 30

• I₀sin² x/x²
• (π/λ)a sin θ

Question 31

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 θ_R = ___. For a disk-shaped aperture of diameter D, θ_R ________.

Answer 31

• (primary) maximum of one coincides with the first zero of the other
• λ/a
• = 1.22 λ/D

Question 32

Light is a form of electromagnetic radiation, with 400 <λ < 700 nm. Light waves are _______ waves, with both _________ E and _________ B components, where E ⊥ B and both E, B perpendicular to the propagation direction of the wave.

Answer 32

• transverse
• electric field
• magnetic field

Question 33

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).

Answer 33

• type of polarisation
• linear
• circular
• elliptical

Question 34

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.

Answer 34

• two orthogonal components of E that are 90^o out of phase

Question 35

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

Answer 35

• θ_p = tan−1 (n_t/n_i)

Question 36

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)

Answer 36

• reflection from a surface

Question 37

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.

Answer 37

• I₀ cos² θ
• intensity of the incident light
• angle of the polarisation

Question 38

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.

Answer 38

• 1/p + 1/q = 1/f
• object distance
• image distance
• focal distance of the lens (or mirror)

Question 39

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.

Question 40

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.

Question 41

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 ___________.

Answer 41

• virtual image

Question 42

• 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.

Answer 42

• Diffraction

Question 47

Approximating the hypergeometric with the Binomial:

Answer 47

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.