---
title: "The Scalar Model of Light"
book: "University Physics — Year 2"
subject: physics
language: en
chapter: 18
exercises: 12
source: https://one-course.com/books/physics/4/en/chapter/18-the-scalar-model-of-light
---

# Chapter 18 — The Scalar Model of Light

Shine two torches on the same wall and the patch is simply twice as bright; shine two beams from one laser on it and the patch breaks into bright and dark fringes. The torches add their intensities, the laser beams add their amplitudes — and every interferometer, hologram and diffraction grating of the coming chapters rests on knowing when light does which. This chapter sets up the tools: light as a *scalar* wave with a phase, the *[optical path](#def-b2-scalar-light-model-path)* that measures that phase along a ray, the way a source emits — in short [wave trains](#prop-b2-scalar-light-model-coherence) whose *[coherence length](#prop-b2-scalar-light-model-coherence)* decides whether two waves can interfere — and the way a detector records it: an average over billions of periods that keeps only the intensity.

## 18.1 Light as a scalar wave

**Definition 18.1 (Scalar model; monochromatic wave).**

In most of wave optics the vector nature of light plays no role (the superposed waves share one polarization, or the light is unpolarized and the polarizations average out): the field is replaced by a real *scalar vibration*

$$
s(M, t) = a(M)\cos\bigl(\omega t - \varphi(M)\bigr) ,
$$

with an amplitude $a(M)$ and a phase $\varphi(M)$ at each point, both slowly varying on the scale of a wavelength. A *monochromatic* wave has a single $\omega$: an idealization, since a real source emits over a band of frequencies. Visible light: $\lambda_0$ from $400$ to $750\,\mathrm{nm}$ in vacuum, $\nu = c/\lambda_0$ from $7.5 \times 10^{14}$ to $4 \times 10^{14}$ Hz, periods of about $2\,\mathrm{fs}$.

**Definition 18.2 (Optical path and phase).**

In a medium of index $n$ the wave travels at $c/n$ and its wavelength is $\lambda_0/n$. The *optical path* along a curve from $A$ to $B$ is

$$
(AB) = \int_A^Bn\,\dd s ,
$$

and the phase accumulated by a [monochromatic wave](#def-b2-scalar-light-model-scalar) travelling along a ray from $A$ to $B$ is

$$
\varphi(B) - \varphi(A) = \frac{2\pi}{\lambda_0}\,(AB) = \omega\,\tau_{AB} ,
$$

$\tau_{AB}$ being the travel time. A *wavefront* is a surface of equal phase; in an isotropic medium the rays of geometrical optics are perpendicular to the wavefronts (Malus–Dupin theorem), and the optical path between two wavefronts is the same along every ray.

**Justification.** The phase advances by $2\pi$ per local wavelength $\lambda_0/n$, i.e. by $2\pi n\,\dd s/\lambda_0$ per element $\dd s$; summed along the ray. The wave $a\cos(\omega t - \varphi)$ is a solution of the [wave equation](https://one-course.com/books/physics/4/en/chapter/6-waves-on-strings-and-rods-the-dalembert-equation#thm-b2-waves-on-strings-equation) in which the energy travels along $\operatorname{\vect{grad}}\varphi$, perpendicular to the surfaces $\varphi =$ const: that is the ray, and the "[optical path](#def-b2-scalar-light-model-path) = same between two [wavefronts](#def-b2-scalar-light-model-path)" statement is the definition of a [wavefront](#def-b2-scalar-light-model-path) read backward. The full theorem (it survives reflections and refractions) is admitted. ∎

**Example 18.3 (Stigmatism; the lens as a phase plate).**

A point source $A$ imaged by a perfect (stigmatic) instrument at $A'$: all the rays from $A$ to $A'$ have the *same [optical path](#def-b2-scalar-light-model-path)* — the waves arriving along every ray are in phase at $A'$, which is why they pile up into a bright point there. For a thin lens of focal length $f$ this is a statement about phase: a plane wave arriving along the axis must leave as a [spherical wave](https://one-course.com/books/physics/4/en/chapter/7-sound-waves-in-fluids#prop-b2-sound-waves-spherical) converging to the focus $F'$; since a [spherical wave](https://one-course.com/books/physics/4/en/chapter/7-sound-waves-in-fluids#prop-b2-sound-waves-spherical) centred at distance $f$ has, at distance $r$ from the axis, the phase $2\pi r^2/2\lambda f$ ahead of its axial value (paraxial approximation, $\sqrt{f^2 + r^2} \approx f + r^2/2f$), the lens must *retard* the wave by $\varphi_L(r) = -\pi r^2/\lambda f$: it is thicker at the centre by exactly the amount that makes all [optical paths](#def-b2-scalar-light-model-path) to $F'$ equal. A lens is a phase plate; a hologram or a liquid-crystal display that imposes the same phase map is a lens.

![Left: a thin lens turns plane wavefronts into spherical ones converging on the focus — all optical paths to F' are equal. Right: inside glass the wavelength shrinks to _0/n and the phase advances faster: the optical path counts the geometric length n times.](https://one-course.com/images/onecourse/chapters/physics-4/b2-scalar-light-model/fig-34d842c7ec6d.svg)

*Left: a thin lens turns plane [wavefronts](#def-b2-scalar-light-model-path) into spherical ones converging on the focus — all [optical paths](#def-b2-scalar-light-model-path) to $F'$ are equal. Right: inside glass the wavelength shrinks to $\lambda_0/n$ and the phase advances faster: the [optical path](#def-b2-scalar-light-model-path) counts the geometric length $n$ times.*

## 18.2 How light is emitted: coherence

**Proposition 18.4 (Wave trains; coherence time and length).**

A classical source (a flame, a lamp, a star) is a collection of atoms each emitting, at random instants and with random phases, short *[wave trains](#prop-b2-scalar-light-model-coherence)* of duration $\tau_c$: the vibration it produces at a point keeps a steady phase only over $\tau_c$ and then jumps. A train of duration $\tau_c$ is not monochromatic: its spectrum spans a frequency band $\Delta\nu \approx 1/\tau_c$ (a Fourier reciprocity, admitted: a finite sinusoid cannot have a single frequency). The *[coherence time](#prop-b2-scalar-light-model-coherence)* $\tau_c$ and the *[coherence length](#prop-b2-scalar-light-model-coherence)*

$$
\ell_c = c\,\tau_c \approx \frac c{\Delta\nu} = \frac{\lambda^2}{\Delta\lambda}
$$

characterize the source: white light $\Delta\lambda \approx 300\,\mathrm{nm}$, $\ell_c
\approx 1\,\text{µ}\mathrm{m}$; a spectral lamp ($\Delta\lambda \sim 0.01\,\mathrm{nm}$), $\ell_c \sim
3\,\mathrm{cm}$; a laser, metres to kilometres. Two waves derived from the same source can interfere only if their path difference is smaller than $\ell_c$ — beyond it they belong to different trains, with unrelated phases.

**Proof.** $\ell_c = c/\Delta\nu$ and $|\Delta\nu/\nu| = |\Delta\lambda/\lambda|$ with $\nu = c/\lambda$ give $\ell_c =
\lambda^2/\Delta\lambda$. The [spectral width](#prop-b2-scalar-light-model-coherence) itself has physical causes: the natural width (radiation damping, [Chapter 17](https://one-course.com/books/physics/4/en/chapter/17-dipole-radiation-and-scattering#ch-b2-dipole-radiation)), the Doppler broadening by the atoms’ thermal motion, collisions — each shortening the train. ∎

![The vibration from a classical source: a succession of wave trains of duration _c with random phase jumps between them; two copies of it can interfere only if their delay is shorter than _c.](https://one-course.com/images/onecourse/chapters/physics-4/b2-scalar-light-model/fig-96f0c5384d58.svg)

*The vibration from a classical source: a succession of [wave trains](#prop-b2-scalar-light-model-coherence) of duration $\tau_c$ with random phase jumps between them; two copies of it can interfere only if their delay is shorter than $\tau_c$.*

**Remark 18.5 (Two sources, one source).**

Two distinct lamps (or two distinct atoms) emit trains with independent random phases: their relative phase changes every $\tau_c
\sim 1 \times 10^{-9}\,\mathrm{s}$, a million times faster than any eye or camera responds — the interference term averages to zero and the intensities add. Interference needs two waves that *remember the same phase jumps*: two copies of one wave, obtained by splitting it (two slits, two mirrors, the two faces of a film) and recombined with a delay shorter than $\tau_c$. A laser, whose trains last microseconds or more, relaxes this constraint enormously; it never removes it.

## 18.3 Detectors and intensity

**Proposition 18.6 (What a detector measures).**

Every light detector — the eye (response time $\sim0.1\,\mathrm{s}$), a photographic film, a CCD pixel ($1\,\mathrm{ms}$), a photodiode ($1\,\mathrm{ns}$) — responds on a time long compared with the period ($10^{-15}$ s) and delivers the *intensity*, the time average of the squared vibration,

$$
I = K\,\langle s^2\rangle = \tfrac12K\,a^2 ,
$$

($K$ a constant, often dropped; $I$ is proportional to the irradiance in $\mathrm{W}/\mathrm{m}^{2}$). The absolute level matters in one respect only: light arrives in photons of energy $h\nu$, and a detector counts on average $I/h\nu$ photons per unit area and time.

**Proof.** $\langle\cos^2(\omega t - \varphi)\rangle = \tfrac12$ over any interval much longer than $2\pi/\omega$. Photons: the last chapter of the Year 1 volume; this is where the scalar wave meets the quantum. ∎

**Example 18.7 (Orders of magnitude).**

A $60\,\mathrm{W}$ bulb radiating $3\,\mathrm{W}$ of visible light at $2\,\mathrm{m}$: $I = 3/4\pi\cdot4 = 60\,\mathrm{mW}/\mathrm{m}^{2}$, $I/h\nu = 1.6 \times 10^{17}$ photons per square metre and second, $3 \times 10^{12}$ per second through a $5\,\mathrm{mm}$ pupil — in the eye’s $0.1\,\mathrm{s}$ there is no trace of the graininess. The faintest star the eye sees sends $\sim 10^3$ photons per second into the pupil; a rod cell fires on a handful. Sunlight: $1\,\mathrm{kW}/\mathrm{m}^{2}$, $3 \times 10^{21}$ photons per square metre and second.

![Time scales of optics: the period of the vibration, the duration of the wave trains of classical and laser sources, and the response times of detectors — all detectors average over many periods, and over many trains of a lamp.](https://one-course.com/images/onecourse/chapters/physics-4/b2-scalar-light-model/fig-15d71751a120.svg)

*Time scales of optics: the period of the vibration, the duration of the [wave trains](#prop-b2-scalar-light-model-coherence) of classical and laser sources, and the response times of detectors — all detectors average over many periods, and over many trains of a lamp.*

**Remark 18.8 (Coherent and incoherent superposition).**

Two vibrations $a_1\cos(\omega t - \varphi_1)$ and $a_2\cos(\omega t - \varphi_2)$ at a point give $s = s_1 + s_2$ and

$$
I = \tfrac12K\bigl[a_1^2 + a_2^2 + 2a_1a_2\langle\cos(\varphi_1 - \varphi_2)\rangle\bigr] .
$$

If the phase difference is stable during the detection (*coherent* waves): $I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos(\varphi_1 - \varphi_2)$, the interference term, which the next chapter studies. If it wanders randomly (*incoherent* waves): $\langle\cos\rangle = 0$ and $I = I_1 +
I_2$ — the torches.

**Method 18.9 (Setting up an optics problem).**

(1) Write each wave as $a\cos(\omega t - \varphi)$ with $\varphi = 2\pi(\text{optical
path})/\lambda_0$ from the source (or from a common [wavefront](#def-b2-scalar-light-model-path)). (2) Decide coherence: same source, path difference $< \ell_c$? Then add amplitudes (complex notation is convenient: $\underline s = a\eu^{\iu(\omega t - \varphi)}$, $I =
\tfrac12K|\underline s|^2$); otherwise add intensities. (3) For lenses and mirrors, use equal [optical paths](#def-b2-scalar-light-model-path) between conjugate points, or the phase-plate picture. (4) Convert to what is measured: intensity, contrast, photon counts, and check the detector’s resolution in time and space.

## 18.4 Exercises

**Exercise 18.1 ★.**

A beam of $600\,\mathrm{nm}$ crosses $1.0\,\mathrm{cm}$ of glass ($n = 1.5$). [Optical path](#def-b2-scalar-light-model-path); number of wavelengths inside; extra phase and extra delay compared with $1\,\mathrm{cm}$ of air; thickness of glass that delays the wave by exactly one wavelength relative to air.

**Solution of Exercise 18.1.**

$(AB) = 1.5\,\mathrm{cm}$; $25\,000$ wavelengths; $0.5\,\mathrm{cm}$ more than air, $8300$ wavelengths, $\Delta\varphi = 2\pi \times 8300$, delay $17\,\mathrm{ps}$; one wavelength of delay for $e = \lambda/(n - 1) = 1.2\,\text{µ}\mathrm{m}$.

**Exercise 18.2 ★.**

[Coherence time](#prop-b2-scalar-light-model-coherence) and length of: white light ($400$–$750\,\mathrm{nm}$); a sodium lamp line of width $0.02\,\mathrm{nm}$ at $589\,\mathrm{nm}$; a red LED ($630\,\mathrm{nm}$, $\Delta\lambda = 25\,\mathrm{nm}$); a multimode laser diode ($\Delta\nu = 100\,\mathrm{GHz}$); a stabilized laser ($\Delta\nu = 1\,\mathrm{kHz}$). Which could give fringes with a path difference of $1\,\mathrm{mm}$? of $10\,\mathrm{m}$?

**Solution of Exercise 18.2.**

$\ell_c = \lambda^2/\Delta\lambda$ or $c/\Delta\nu$, $\tau_c = \ell_c/c$: white $0.9\,\text{µ}\mathrm{m}$ ($3\,\mathrm{fs}$); sodium $1.7\,\mathrm{cm}$ ($60\,\mathrm{ps}$); LED $16\,\text{µ}\mathrm{m}$ ($50\,\mathrm{fs}$); diode $3\,\mathrm{mm}$ ($10\,\mathrm{ps}$); stabilized $300\,\mathrm{km}$ ($1\,\mathrm{ms}$). $1\,\mathrm{mm}$: the sodium lamp, the diode and the laser; $10\,\mathrm{m}$: the stabilized laser only.

**Exercise 18.3 ★.**

A $1\,\mathrm{mW}$ laser pointer at $650\,\mathrm{nm}$: photons per second; a $100\,\mathrm{W}$ bulb ($5\,\mathrm{W}$ visible, mean $550\,\mathrm{nm}$) at $3\,\mathrm{m}$: photons per second through a $6\,\mathrm{mm}$ pupil; number during the eye’s $0.1\,\mathrm{s}$; relative fluctuation $1/\sqrt N$ of that number — is the light grainy?

**Solution of Exercise 18.3.**

$10^{-3}/3.06 \times 10^{-19} = 3.3 \times 10^{15}\,$ per second. Bulb: $I = 5/4\pi \times 9 =
44\,\mathrm{mW}/\mathrm{m}^{2}$, pupil $28\,\mathrm{mm}^{2}$: $1.2\,\text{µ}\mathrm{W}$, $3.5 \times 10^{12}$ photons per second, $3.5 \times 10^{11}$ in $0.1\,\mathrm{s}$, fluctuation $2 \times 10^{-6}$: perfectly smooth.

**Exercise 18.4 ★.**

A point source at $1.0\,\mathrm{m}$ from a screen, $\lambda = 500\,\mathrm{nm}$. [Optical path](#def-b2-scalar-light-model-path) difference between the centre of the screen and a point $1\,\mathrm{cm}$ off axis (exactly, then with the paraxial formula $r^2/2d$); number of wavelengths; radius of the [wavefront](#def-b2-scalar-light-model-path) at the screen; at what off-axis distance does the paraxial formula err by one wavelength?

**Solution of Exercise 18.4.**

Exact $\sqrt{1 + 10^{-4}} - 1 = 49.9988\,\text{µ}\mathrm{m}$, paraxial $50\,\text{µ}\mathrm{m}$: a hundred wavelengths; the [wavefront](#def-b2-scalar-light-model-path) is a sphere of radius $1\,\mathrm{m}$; the quartic term $r^4/8d^3$ reaches $\lambda$ at $r = (8\lambda d^3)^{1/4} = 4.5\,\mathrm{cm}$.

**Exercise 18.5 ★★.**

*The lens as a phase plate.* (a) A plane wave along the axis has a uniform phase on the plane of a thin lens; after the lens the phase is $\varphi_L(r) = -\pi r^2/\lambda f$ relative to the centre. Show that the emerging [wavefront](#def-b2-scalar-light-model-path) is, in the paraxial approximation, a sphere of radius $f$ centred on $F'$. (b) A point source at distance $d$ before the lens sends a wave whose phase on the lens plane is $+\pi r^2/\lambda d$ (relative to the centre): show that after the lens the wave converges at $d'$ with $1/d' = 1/f - 1/d$, the lens formula. (c) Thickness profile of a plano-convex glass lens ($n = 1.5$) of focal length $10\,\mathrm{cm}$: $e(r) =
e_0 - r^2/2(n - 1)f$; sag at $r = 1\,\mathrm{cm}$. (d) A "Fresnel lens" folds the profile back every time it exceeds $\lambda/(n - 1)$: why does it still focus, and why only one colour perfectly?

**Solution of Exercise 18.5.**

(a) A sphere of radius $f$ centred at $F'$ has, on the lens plane, the phase $-\pi r^2/\lambda f$ relative to the axis (paraxially): exactly the lens’s. (b) Total phase $\pi r^2/\lambda\,(1/d - 1/f) = -\pi r^2/\lambda d'$: a wave converging at $d'$ with $1/d' = 1/f - 1/d$. (c) The glass adds $(n - 1)e(r)
\times2\pi/\lambda$: $e(r) = e_0 - r^2/2(n - 1)f$; sag $10^{-4}/0.1 = 1\,\mathrm{mm}$. (d) A phase defined modulo $2\pi$ is the same phase — at the design wavelength; at others the steps are no longer whole wavelengths.

**Exercise 18.6 ★★.**

*Stigmatism of a mirror.* A mirror must send a plane wave arriving along its axis to a point $F$. (a) Write the condition "equal [optical paths](#def-b2-scalar-light-model-path) from a plane [wavefront](#def-b2-scalar-light-model-path) to $F$ for every point of the mirror" and show it defines a parabola $y^2 = 4fx$ with $F$ at its focus. (b) Why is a spherical mirror only approximately stigmatic (compare $y^2 = 4fx$ with the circle of radius $2f$ near the vertex)? (c) For a $20\,\mathrm{cm}$ mirror of $f = 1\,\mathrm{m}$, the path error of the sphere at the edge, in wavelengths of $500\,\mathrm{nm}$. (d) Why do radio telescopes tolerate sphericity that optical ones cannot?

**Solution of Exercise 18.6.**

(a) From a plane [wavefront](#def-b2-scalar-light-model-path) at $x = -D$ to $(x, y)$ then to $F(f, 0)$: $D + x
+ \sqrt{(x - f)^2 + y^2} = D + f$, so $\sqrt{(x - f)^2 + y^2} = f - x$, $y^2 = 4fx$, with $F$ at the focus. (b) The circle $x = 2f - \sqrt{4f^2 - y^2} \approx y^2/4f +
y^4/64f^3$ agrees to second order only. (c) $y^4/64f^3 = 1.6\,\text{µ}\mathrm{m}$, doubled by reflection: $3\,\text{µ}\mathrm{m}$, six wavelengths — hence parabolic mirrors. (d) At $\lambda \sim 1\,\mathrm{cm}$ the same error is a thousandth of a wavelength.

**Exercise 18.7 ★★.**

*Fermat.* The [optical path](#def-b2-scalar-light-model-path) from $A$ (in medium $n_1$) to $B$ (in $n_2$) through a point $P$ of the plane interface is $n_1AP + n_2PB$. (a) Show that it is stationary (minimal) for the point $P$ satisfying $n_1\sin\theta_1 = n_2\sin\theta_2$. (b) Interpret: neighbouring paths have the same phase, so the waves along them add — this is why light "chooses" the Descartes ray. (c) Derive the law of reflection the same way. (d) Why is the path through a lens from $A$ to its image $A'$ the same along every ray, and not merely stationary?

**Solution of Exercise 18.7.**

(a) $L(x) = n_1\sqrt{a^2 + x^2} + n_2\sqrt{b^2 + (d - x)^2}$: $L' = n_1\sin\theta_1 -
n_2\sin\theta_2 = 0$. (b) Near the stationary path neighbouring paths have equal phases and their waves add; elsewhere they cancel. (c) $n_1 = n_2$: $\sin\theta_1 = \sin\theta_r$. (d) Stigmatism means all rays from $A$ to $A'$ have equal paths: a whole family of stationary paths, not one.

**Exercise 18.8 ★★.**

*Why two lamps do not interfere.* Two sodium lamps illuminate a screen; at a point, their waves have amplitudes $a$ and a relative phase $\Delta\varphi$ that jumps randomly every $\tau_c = 1 \times 10^{-10}\,\mathrm{s}$. (a) Intensity at that point as a function of $\Delta\varphi$; its maximum and minimum. (b) The detector averages over $\tau_d = 1\,\mathrm{ms}$: how many independent values of $\Delta\varphi$? Mean intensity; relative size of the residual fluctuation of the interference term ($\sim 1/\sqrt N$). (c) A photodiode with $\tau_d = 1 \times 10^{-11}\,\mathrm{s}$: what would it see? (d) How did the first experiment showing interference between two independent lasers (1963) manage it?

**Solution of Exercise 18.8.**

(a) $I = 2I_0(1 + \cos\Delta\varphi)$: $4I_0$ and $0$. (b) $N = 10^7$: mean $2I_0$, residual $\sim 2I_0/\sqrt N = 6 \times 10^{-4}$ of it. (c) During $10\,\mathrm{ps}$ the phase is frozen: it would see fringes, jumping every $0.1\,\mathrm{ns}$. (d) Lasers with $\tau_c \sim 1\,\text{µ}\mathrm{s}$ and a detector faster than that: the fringes exist for a microsecond and were photographed.

**Exercise 18.9 ★★.**

*Apparent depth.* A coin lies at depth $h$ in water ($n = 1.33$). (a) Using the [optical path](#def-b2-scalar-light-model-path) (or Descartes’ law at small angles), show that an eye above sees it at depth $h/n$. (b) $h = 1.2\,\mathrm{m}$: apparent depth; by how much does the [optical path](#def-b2-scalar-light-model-path) from the coin to the eye exceed its geometric length? (c) A fish at depth $h$ sees a fly $1\,\mathrm{m}$ above the surface at what apparent height? (d) Why does a pool look shallower still when viewed obliquely?

**Solution of Exercise 18.9.**

(a) Small angles: $\tan i \approx n\tan r$, the rays appear to come from $h/n$. (b) $0.90\,\mathrm{m}$; the [optical path](#def-b2-scalar-light-model-path) is $n \times 1.2 = 1.6\,\mathrm{m}$, $0.4\,\mathrm{m}$ more. (c) $n \times 1 = 1.33\,\mathrm{m}$. (d) At large angles the refraction is stronger and the apparent depth smaller.

**Exercise 18.10 ★★★.**

*Contrast and coherence.* Two copies of a wave with a path difference $\delta$ are superposed; the source’s spectrum is spread uniformly over $[\nu_0 - \Delta\nu/2, \nu_0 + \Delta\nu/2]$, each frequency interfering with itself only. (a) Intensity from one frequency: $2I_0
[1 + \cos(2\pi\nu\delta/c)]$. (b) Integrate over the band and show that the total is $2I_0[1 + \operatorname{sinc}(\pi\Delta\nu\,\delta/c)\cos(2\pi\nu_0\delta/c)]$ with $\operatorname{sinc}x = \sin x/x$. (c) Define the contrast $V = (I_{\max} -
I_{\min})/(I_{\max} + I_{\min})$ of the fringes near $\delta$ and show $V =
|\operatorname{sinc}(\pi\Delta\nu\,\delta/c)|$: it vanishes at $\delta = c/\Delta\nu = \ell_c$. (d) Numbers for a LED ($\Delta\lambda = 25\,\mathrm{nm}$ at $630\,\mathrm{nm}$): $\delta$ at which the contrast first vanishes.

**Solution of Exercise 18.10.**

(a) Two equal waves with phase difference $2\pi\nu\delta/c$. (b) Average over the band:

$$
\frac1{\Delta\nu}\int\cos\frac{2\pi\nu\delta}c\,\dd\nu = \cos\frac{2\pi\nu_0\delta}c\,\operatorname{sinc}\frac{\pi\Delta\nu\,\delta}c .
$$

(c) $I_{\max, \min} = 2I_0(1 \pm |\operatorname{sinc}|)$: $V = |\operatorname{sinc}(\pi\Delta\nu\delta/c)|$, zero at $\delta = c/\Delta\nu$. (d) $\Delta\nu = c\Delta\lambda/\lambda^2 = 1.9 \times 10^{13}\,\mathrm{Hz}$: $\delta =
16\,\text{µ}\mathrm{m}$.

**Exercise 18.11 ★★★.**

*Seeing stars.* The eye detects a flash when about $10$ photons reach a rod within $0.1\,\mathrm{s}$; the dark-adapted pupil is $7\,\mathrm{mm}$; take $500\,\mathrm{nm}$. (a) Minimum [photon flux](#prop-b2-scalar-light-model-intensity) and irradiance at the eye. (b) The Sun gives $1\,\mathrm{kW}/\mathrm{m}^{2}$; a star of magnitude $m$ gives $10^{-0.4(m + 26.7)}$ of it: irradiance of a magnitude-6 star (the naked-eye limit) and the photon rate into the pupil; consistent with (a)? (c) A telescope of $200\,\mathrm{mm}$ aperture: gain in photons, and the magnitude it reaches. (d) Why do astronomers speak of the "photon noise" of a faint image, and how does it scale with exposure time?

**Solution of Exercise 18.11.**

(a) $100$ photons per second on $38\,\mathrm{mm}^{2}$: $2.6 \times 10^{6}\,\mathrm{m}^{-2}\,\mathrm{s}^{-1}$, $\times4 \times
10^{-19}$ J: $1 \times 10^{-12}\,\mathrm{W}/\mathrm{m}^{2}$. (b) $10^{-13.1} \times 1000 = 8 \times 10^{-11}\,\mathrm{W}/\mathrm{m}^{2}$, $2 \times 10^8$ photons per square metre and second, $8000$ per second into the pupil — well above (a): the eye’s real limit is set by background and by the spread over many rods. (c) $(200/7)^2 = 800$, $+7.3$ magnitudes: $m \approx
13$. (d) $N$ photons fluctuate by $\sqrt N$: the signal-to-noise grows as the square root of the exposure.

**Exercise 18.12 ★★★.**

*Beyond paraxial.* A [spherical wave](https://one-course.com/books/physics/4/en/chapter/7-sound-waves-in-fluids#prop-b2-sound-waves-spherical) from a point at distance $d$ has, at the transverse distance $r$, the exact extra path $\sqrt{d^2 +
r^2} - d$. (a) Expand to fourth order: $r^2/2d - r^4/8d^3$. (b) The quadratic (paraxial) term is kept and the quartic dropped when the latter is below $\lambda/4$: show this requires $r^4 < 2\lambda d^3$. (c) Numbers: $d = 1\,\mathrm{m}$, $\lambda = 500\,\mathrm{nm}$: maximum $r$; $d = 1\,\mathrm{cm}$ (a microscope): maximum $r$, and the corresponding angle. (d) Comment: the "aberrations" of geometrical optics are these neglected terms.

**Solution of Exercise 18.12.**

(a) $d(1 + r^2/2d^2 - r^4/8d^4 + \dots) - d$. (b) $r^4/8d^3 < \lambda/4$. (c) $r <
(2\lambda d^3)^{1/4}$: $3.2\,\mathrm{cm}$ for $1\,\mathrm{m}$; $1\,\mathrm{mm}$ for $1\,\mathrm{cm}$, an angle of $0.1\,\mathrm{rad}$. (d) Spherical aberration is this quartic term; the other aberrations are its off-axis cousins.

![Low-pressure sodium lamps in fog: a single yellow line, a coherence length of centimetres — the classic source of the optics laboratory, and of the old streets.](https://one-course.com/images/onecourse/chapters/physics-4/b2-scalar-light-model/img-9aab3b539316.jpg)

*Low-pressure sodium lamps in fog: a single yellow line, a [coherence length](#prop-b2-scalar-light-model-coherence) of centimetres — the classic source of the optics laboratory, and of the old streets.*

## 18.5 Problem: A sodium lamp and a laser pointer

**Problem 18.1.**

Weekend problem — two light sources taken apart: the yellow lamp of the old street, the red pointer in the lecture hall, and the one number that decides whether each can make fringes

**Part I — The sodium lamp.** Sodium emits its yellow doublet at $\lambda_1 = 589.0\,\mathrm{nm}$ and $\lambda_2 =
589.6\,\mathrm{nm}$. The natural width of each line is $\Delta\nu_{\text{nat}} =
10\,\mathrm{MHz}$; in the lamp the atoms are at $500\,\mathrm{K}$ (sodium mass $23\,\mathrm{u}$, $k_B = 1.38 \times 10^{-23}\,\mathrm{J}/\mathrm{K}$, $u = 1.66 \times 10^{-27}\,\mathrm{kg}$).

1. Frequencies of the two lines and their difference.
2. [Coherence length](#prop-b2-scalar-light-model-coherence) of a single line if only its natural width counted.
3. Doppler broadening: an atom moving at $v$ along the line of sight emits at $\nu(1 + v/c)$ ; with the thermal rms speed $\sqrt{k_BT  /m}$ along one axis, estimate $\Delta\nu_D$ and compare with the natural width.
4. [Coherence length](#prop-b2-scalar-light-model-coherence) of one Doppler-broadened line; in a real high-pressure lamp collisions broaden the line a further ten times: [coherence length](#prop-b2-scalar-light-model-coherence) then.
5. The two lines together: over what path difference do their fringe systems go from coincidence to opposition and back (the "beat length" $\lambda^2/\Delta\lambda$ )? How many fringes is that?
6. A student makes a two-beam interferometer with the lamp and increases the path difference from zero: describe the contrast seen — the beats and the final extinction — with the numbers found.
7. The lamp radiates $1\,\mathrm{W}$ of yellow light: photons per second; energy of one photon in eV, and why the light is yellow.
8. Two sodium lamps side by side: can their lights interfere? Why is the question different for the two lines of one lamp?

**Part II — The laser pointer.** A $1\,\mathrm{mW}$ red diode laser at $650\,\mathrm{nm}$ emits, when cheap, several longitudinal modes spread over $\Delta\lambda = 0.5\,\mathrm{nm}$; a stabilized single-mode one has $\Delta\nu = 1\,\mathrm{MHz}$.

9. [Coherence lengths](#prop-b2-scalar-light-model-coherence) of the two pointers.
10. Photons emitted per second; mean spacing between photons along the beam (in metres); compare with the [coherence length](#prop-b2-scalar-light-model-coherence) — what does that say about "photons interfering"?
11. Convert the $0.5\,\mathrm{nm}$ spread into a frequency width; the diode’s cavity is $1\,\mathrm{mm}$ long with $n = 3.5$ : spacing of its longitudinal modes ( $c/2nL$ ) and the number of modes in the spread.
12. The beam ( $1\,\mathrm{mm}$ diameter) falls on a photodiode whose every photon yields an electron: current.
13. The cheap pointer is used in a two-beam interferometer with a $2\,\mathrm{mm}$ path difference: fringes or not? And the stabilized one with $10\,\mathrm{m}$ ?
14. The diode’s output is in fact switched on and off in $1\,\mathrm{ns}$ pulses: can the pulses from two successive periods interfere with each other? What sets the coherence here, the pulse length or the line width?
15. Why does a laser’s light look "speckled" on a wall, and why does the lamp’s not?
16. A red LED at $650\,\mathrm{nm}$ has $\Delta\lambda = 25\,\mathrm{nm}$ : [coherence length](#prop-b2-scalar-light-model-coherence) ; why do some projectors prefer LEDs to lasers?

**Part III — [Optical paths](#def-b2-scalar-light-model-path) and detectors.**

17. A $2\,\mathrm{mm}$ glass slide ( $n = 1.5$ ) is inserted in one arm of the interferometer: extra [optical path](#def-b2-scalar-light-model-path) , and the number of fringes that shift past a mark.
18. The slide is tilted by $10{}^{\circ}$ : extra path (use $e/\cos r$ for the length in the glass, minus the air it replaces, to first approximation $e(n/\cos r - \cos(i - r)/\cos r)$ — or simply estimate with the longer geometric path); order of magnitude of the fringe shift.
19. The detector is a camera with $10\,\text{µ}\mathrm{m}$ pixels; the fringes are $50\,\text{µ}\mathrm{m}$ apart: how many pixels per fringe, and what happens to the measured contrast if the spacing falls to $10\,\text{µ}\mathrm{m}$ ?
20. Exposure $10\,\mathrm{ms}$ at $1\,\text{µ}\mathrm{W}/\mathrm{cm}^{2}$ on a pixel: photons per pixel; relative photon noise.
21. Two equal coherent waves of intensity $I_0$ meet with a path difference $\delta$ : write the intensity $I(\delta)$ and the period of the fringes in $\delta$ .
22. The fringes drift past a point at $1\,\mathrm{kHz}$ (a vibrating mirror): what does the eye see, what does the photodiode see?
23. The interferometer’s arms run through $10\,\mathrm{m}$ of air ( $n - 1 =  2.9 \times 10^{-4}\,$ ): [optical path](#def-b2-scalar-light-model-path) in excess of vacuum, in fringes; what a $1\,\%$ change of air pressure does to the pattern.
24. The lamp and the pointer give the same $1\,\text{µ}\mathrm{W}/\mathrm{cm}^{2}$ : which gives more photons per second, and why does it not matter for the fringes?
25. Sum up: for each source, the [coherence length](#prop-b2-scalar-light-model-coherence) , what limits it, and the interferometer path difference it allows.

**Solution of Problem 18.1.**

**1.** $\nu_1 = 5.0934 \times 10^{14}\,\mathrm{Hz}$, $\nu_2 = 5.0882 \times 10^{14}\,\mathrm{Hz}$: $\Delta\nu = 5.2 \times 10^{11}\,\mathrm{Hz}$.

**2.** $c/\Delta\nu_{\text{nat}} = 30\,\mathrm{m}$.

**3.** $\sqrt{k_BT/m} = 425\,\mathrm{m}/\mathrm{s}$: $\Delta\nu_D \approx 2\nu v/c \approx 1.4\,\mathrm{GHz}$, a hundred times the natural width.

**4.** $c/\Delta\nu_D \approx 20\,\mathrm{cm}$; with collisions, about $2\,\mathrm{cm}$.

**5.** $\lambda^2/\Delta\lambda = 589^2/0.6$ nm $= 0.58\,\mathrm{mm}$: coincidence to opposition every $0.29\,\mathrm{mm}$; about $1000$ fringes per beat.

**6.** The contrast oscillates with a period of $0.58\,\mathrm{mm}$ in path difference (zero at $0.29$, $0.87$, $1.45\,\mathrm{mm}$, …) inside an envelope that fades over a few centimetres: some fifty beats, then nothing.

**7.** $3 \times 10^{18}\,$ photons per second; $2.1\,\mathrm{eV}$, the yellow of the spectrum.

**8.** No: independent atoms, random phases. The two lines of one lamp come from different atoms too and do not interfere with each other: the beats are two fringe systems adding in *intensity*.

**9.** $\lambda^2/\Delta\lambda = 0.85\,\mathrm{mm}$; $c/\Delta\nu = 300\,\mathrm{m}$.

**10.** $3.3 \times 10^{15}\,$ per second, spaced $c/N = 90\,\mathrm{nm}$ along the beam: ten thousand photons within a [coherence length](#prop-b2-scalar-light-model-coherence). Interference is not photons meeting: each photon interferes with itself.

**11.** $\Delta\nu = c\Delta\lambda/\lambda^2 = 360\,\mathrm{GHz}$; mode spacing $c/2nL = 43\,\mathrm{GHz}$: about eight modes.

**12.** $e \times 3.3 \times 10^{15} = 0.5\,\mathrm{mA}$.

**13.** $2\,\mathrm{mm}$ $>$ $0.85\,\mathrm{mm}$: no fringes; $10\,\mathrm{m}$ $<$ $300\,\mathrm{m}$: fringes.

**14.** A nanosecond pulse is $30\,\mathrm{cm}$ long, far longer than the $0.85\,\mathrm{mm}$ [coherence length](#prop-b2-scalar-light-model-coherence): the line width, not the pulse, sets the coherence; successive pulses interfere only if the laser’s phase survives the off period — it does not.

**15.** Coherent light scattered by the rough wall interferes at the retina with random path differences: speckle. The lamp’s micrometre-scale coherence averages all that out.

**16.** $\lambda^2/\Delta\lambda = 17\,\text{µ}\mathrm{m}$: far too short for speckle on a rough wall — no speckle, a smoother image.

**17.** $(n - 1)e = 1\,\mathrm{mm}$: $1540$ fringes.

**18.** $r = 6.65{}^{\circ}$; extra path $\approx e(n/\cos r - \cos(i - r)/\cos r)
= 3.021 - 2.010 = 1.011\,\mathrm{mm}$, $11\,\text{µ}\mathrm{m}$ more than untilted: some twenty fringes more.

**19.** Five pixels per fringe; at one pixel per fringe the pixel integrates a whole period and the contrast collapses.

**20.** $1 \times 10^{-12}\,\mathrm{W}$ on the pixel $\times$ $10\,\mathrm{ms}$ $= 1 \times 10^{-14}\,\mathrm{J}$: $3 \times 10^4$ photons, noise $0.6\%$.

**21.** $I = 2I_0[1 + \cos(2\pi\delta/\lambda)]$: one fringe per wavelength of path difference.

**22.** The eye averages the kilohertz motion into a uniform field; the photodiode follows it.

**23.** $(n - 1) \times 10 = 2.9\,\mathrm{mm}$: $4500$ fringes; a $1\%$ change of pressure moves them by $45$ fringes — the interferometer is a barometer unless evacuated.

**24.** Nearly the same photon rate ($2.1\,\mathrm{eV}$ against $1.9\,\mathrm{eV}$); the fringes depend on amplitudes and phases, the photon count only on the noise.

**25.** Lamp: centimetres, limited by Doppler and collision broadening, fringes up to about $1\,\mathrm{cm}$; cheap pointer: $0.85\,\mathrm{mm}$, limited by its several modes; stabilized laser: $300\,\mathrm{m}$, limited by its line width.