A note on Feynmanʼs calculation ofreflection amplitudes for radiation striking aglass surface

Giancarlo Reali

Laboratorio Sorgenti Laser, Università di Pavia, via Ferrata 3, 27100 Pavia, Italia

E-mail: giancarlo.reali@unipv.it

Received 8 April 2014, revised 23 April 2014Accepted for publication 8 May 2014Published 5 June 2014

AbstractIn this paper we present a detailed calculation of reflection amplitudes for s-and p-polarized radiation striking a glass surface, closely following the deri-vation found in the Feynman Lectures on Physics, vol I. The basic ideaunderlying Feynmanʼs exposition is the extinction theorem, which is used herein a very unique, Feynmanesque way. The calculation is carried out both forthe case of radiation coming from the air and from the glass. We also showthat the same reasonings are useful to discuss the internal Brewsterʼs law.

Keywords: optics, reflection coefficients, extinction theorem, Brewsterʼs angle

(Some figures may appear in colour only in the online journal)

1. Introduction

Maxwellʼs equations provide a quick and elegant account of why light, entering a transparentmedium (like a plane, thick piece of glass, as in figure 1), propagates with a speed reduced bya factor of n (the index of refraction of the glass), and allows one to find the reflected andrefracted fields by matching the boundary conditions at the vacuum-glass interface.

However, this standard analysis does little to clarify the physical mechanism involved at themicroscopic level. For this deeper explanation, the physical principle that we should have in mindwhen a light wave impinges onto a glass surface from the air (which we assimilate to the vacuumfrom now on) is the following. The glass is made of an assembly of polarizable atoms in vacuum,each of them being set into oscillation when interacting with a radiation field and re-radiating newelectric and magnetic fields into the interatomic vacuum. The response of the glass to theimpinging radiation is then associated with the scattering of light by the atoms.

At a more elementary level, and as far as the interaction of an electromagnetic wave withthe glass is concerned, each atom can be thought of as a classical charged harmonic oscillator

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(an oscillating dipole) driven by the external field and by the dipole fields of all the otheroscillating charges in the glass [1, 2].

In the case of a continuous light beam impinging onto the glass, the resulting total field ata point includes contributions of the incoming primary wave and of the secondary waves re-scattered by all the dipoles set into oscillation by the same total field (the re-radiation from thedipoles contributes to the net electric field driving the dipoles: this is the way the dipolarinteraction is taken into account self-consistently [3, 4]).

Experimentally, this field takes the form of refracted and reflected beams, both containedin the incidence plane of the incoming beam. Inside the glass, the refracted beam is along adirection at a different angle with respect to the direction of the incoming beam in air (exceptfor the case of normal incidence onto the glass surface), and it turns out to propagate with aspeed v = c/n, reduced with respect to c, the speed of light in vacuum, by the index of therefraction n of the glass. Outside the glass, the reflected beam forms an angle equal to theincidence angle but of opposite sign with respect to the normal to the surface, and propagateswith speed c (the reflection is new light re-radiated along its direction by all the dipoles in theglass and not the old light bounced off the glass surface [3] as it seems to happen in figure 1).The index of refraction is the macroscopic parameter that describes the glass response to theincoming radiation and depends on the basic atomic quantities of the glass (electron chargeand mass, resonance frequencies, atomic densities) and also on the frequency of the radiation.It also enters the Snellʼs law that explains the geometrical relations of experiments such asthat exemplified in figure 1.

This scenario, which connects the reflected and refracted fields to the incident field and tothe scattered light by the oscillating dipoles in the glass, from a mathematical point of view isequivalent but more complex than the Maxwellʼs equations and their associated boundaryconditions, since it translates into integral equations [5–7]1 instead of partial differential

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Figure 1. Experiment of a light beam striking an air-glass interface obliquely. Incident,reflected and refracted beams are shown. The dotted line would be the direction of theincident wave in glass if it could go straight into it.

1 In [7], as in references [5, 6], the medium is supposed to be homogeneous, isotropic, nonmagnetic, and to respondlinearly to the impinging field. This is generally true for glass.

equations. Integral equations are more difficult to solve, but have the advantage of notrequiring boundary conditions since they are built into them. It is also possible to put theintegral equations into a form that reinterprets the internal dipole scattering as only due to theaction of surface scatterers at the air-glass interface. This interpretation was already pointedout to be a mathematical artifact [7].

In Volume I of Feynman Lectures on Physics [8], probably the most original of thefamous three-volume set, Feynman presents a calculation of the amplitudes of reflected lightfrom a glass surface, considering a beam of linear polarized light striking the surface where itis partly reflected and partly refracted into the glass. In his exposition it is clear that Feynmanhas in mind the atomic hypothesis of glass made of polarizable atoms in vacuum, where lightpropagates with speed c, as described above. Anyway, he approaches the problem withoutusing any sophisticated mathematical formalism and only elaborates on the geometrical dataavailable from figure 1 to arrive at the final answer. While the main steps of the calculation arecarried out by Feynman, others are only suggested, and the final results for the reflectionamplitudes of both s- and p-polarization are provided without a detailed derivation.

In this paper we shall closely follow the discussion and the notation of Feynman, withoutskipping any of the intermediate steps, which are by themselves physically interesting.Feynmanʼs reasoning will also be used to derive and explain the internal Brewsterʼs law, veryoften the subject of ambiguous discussions on the role of radiating dipoles in the glass [9–11].

2. Absolute squares of reflection amplitudes

Figure 2 describes the two possible cases of an incident electric vector linearly polarized in (a)perpendicular to the plane of incidence (s-polarization), which coincides with the plane of thepaper, and in (b) in the plane of incidence (p-polarization) in the direction of the dotted arrow.

The amplitude of the incident beam is set to 1 for both cases, while the reflected andrefracted amplitudes are b and a for the s-polarized waves, and B and A for the p-polarizedwaves, respectively. The incident and reflected angles are i, while the refracted angle is r.

Feynmanʼs program is simple and general. He uses, without naming it, one of the mostprofound principles of optics, the so-called extinction theorem [5–7]. Indeed, with a flash ofhis magic physical intuition, he runs the argument backwards, starting from the experimental

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Figure 2. Incident wave at an air-glass interface: (a) s-polarization, (b) p-polarization.

observation (figure1) to get correct information about which fields are generated by theincident wave. The principle that we must understand is as follows.

All the sources in the Universe make the net field, which is the incident and the reflectedwave in the air, and the refracted wave in the glass. The source of the incident light beamproduces a field of unit amplitude, which would move into the glass along the dotted lines infigure 2. The polarization current, induced by the total field in the glass, must first produce therefracted wave. In addition, Feynman guesses that the polarization current must also producetwo vacuum waves, which propagate with speed c.

Radiating into the vacuum between the atoms of the glass, the oscillating dipoles gen-erate a first vacuum wave in the air, the reflected beam, that propagates in the plane ofincidence along a direction symmetric to the heavy dotted line shown in figure 2(a) and (b)with respect to the air-glass interface.

The oscillating dipoles also generate another vacuum wave in the glass. If the chargedharmonic oscillators were frozen and could not respond to the driving fields, the incidentbeam would continue to propagate with speed c along the dotted straight line in the vacuumbetween them. This field is not observed, and therefore the oscillating dipoles must alsoproduce a second vacuum field, of amplitude −1, overlapping and cancelling the incidentwave inside the glass. This is the extinction theorem.

For s-polarization, since the fields are all parallel, both −1 and b amplitudes are pro-portional to a, which is parallel to the polarization current. For p-polarization the situation isdifferent, since the fields are all in the plane of incidence and form different angles with oneanother. Since the refracted amplitude A is parallel to the polarization current in this case,Feynman guesses that not all of the field A contributes to generate the amplitudes −1 and B.Only the projections of the A field onto the perpendicular to the directions of propagation of−1 and B in figure 2(b) are effective in generating them. (It can help to remember that for asingle oscillating charge, only the component of its acceleration perpendicular to the line ofsight taken from the observation point generates a radiation field in it.)

Simple geometrical considerations thus lead to the conclusion that B is proportional to+( )Acos i r and that−1 is proportional to −( )Acos i r . Therefore, the following proportions

hold true [12]2:

= +( )ba

BAcos i r

(1)

and

− = −−

( )Acos i r

a11

. (2)

Using equation (2) to get A a and substituting into equation (1), we obtain the ratio B b:

= +−

( )( )

Bb

cos i r

cos i r. (3)

At this intermediate point, Feynman suggests checking this formula against the knownresult of Brewsterʼs angle: when π+ =( )i r 2 this ratio is zero, and he concludes that atleast so far equation (3) is not obviously wrong.

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2 This paper clarifies that the signs of the coefficients are always to be considered in relation with the appropriatediagram of the electric vectors. This connection is extablished by choosing the sign in the projection to the right-hand side of equation (1) to get Feynmanʼs b and B.

Having assumed unit amplitudes for the incident waves, B 12 2 and b 12 2 represent thereflection coefficients for the p- and s-polarized waves, respectively, and their ratio isdetermined by the previously derived formula. However, using conservation of energy, it ispossible to obtain each coefficient B 2 and b 2 individually.

The energy in each wave must be proportional to the squared modulus of the corre-sponding amplitude. However, whichever coefficient we include in front of the squaredmodulus of the amplitudes, it will be the same for the incident and reflected waves, since bothare in the same medium. For that reason the energy in the refracted beam is simply pro-portional either to − B1 2 or − b1 2 for the two polarizations. Thus, energy that passes intothe glass in figure 2(b) is to the energy that passes into the glass in figure 2(a) as the ratio ofthe absolute squares of the refracted amplitudes A a2 2.

Therefore

−− =B

bAa

11

(4)2

2

2

2

and using the previous results to eliminate A a2 2 and to express B in terms of b, we get anexpression that only depends on b 2,

−

− = −

+−

( )

b

b cos i r

1

11

(5)

( )( )

cos i r

cos i r

2

2 2

2

2

which, when solved for b 2, gives

= −+

( )( )

bsin i r

sin i r. (6)

22

2

Then, using the expression for B b, also B 2 is derived:

= −+

( )( )

Btan i r

tan i r. (7)

22

2

Essentially, this is the point where Feynmanʼs computation ends, not without havingsuggested hints on how to find that b is real and how it is also possible to determine its sign.

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Figure 3. Stokes construction: (a) forward-in-time and (b) backward-in-timepropagations.

3. Computation of the amplitudes

To proceed further, let us consider a situation first analysed by Stokes [13] and illustrated infigure 3. In this discussion the polarization is not important, and the result is com-pletely general.

The situation in figure 3(a) is the same as that shown in figure 2. In figure 3(b) we analysewhat we should expect if, at a certain time after the incident field has struck the glass surface,the reflected and refracted waves were time-reversed, letting them exactly retrace their pathsbackwards. Since the light sees different situations depending on which side it is approachingthe glass surface, either from the air or from the glass, to analyse the time-reversed propa-gation we introduce different (at least in principle) reflected and refracted amplitudes, ′b and′a , respectively, and require that, at each time, the reflected and refracted backward-in-timefields should reproduce the amplitudes of forward-in-time fields:

= + ′b a a1 (8)2

= + ′ab b a0 . (9)

Equations (8) and (9) are sometimes referred to as Stokes’ relations. The first relation saysthat, travelling backwards in time, the fields must recompose at the interface to give back theoriginal incident wave of amplitude 1 in air. The second relation accounts for the destructiveinterference that refracted and reflected time-reversed fields set on in glass in order to cancelthe dotted field not present in the forward-in-time propagation.

Since we already know b 2 and that ′ = −b b from equation (9), it is easy to make b realby an appropriate choice of its phase. Setting = ϕb b ei and ′′ = ϕb b ei , we see that the choiceϕ ϕ π′ = + verifies ′ = −b b, and we can take advantage of the arbitrariness of the phase ofb to further choose ϕ π= = ± ±n n, 0, 1, 2 ,..., that results in a real b, determined to within asign:

= ± −+

( )( )

bsin i r

sin i r( ) . (10)

In order to find the sign, we now proceed, as suggested by Feynman, by considering thecase of radiation normally incident on a very thin, plane parallel glass layer, of thicknessΔ λ< <z , the vacuum wavelength of the incident wave. Taking the limit of small i and rangles in equation (10), and using Snellʼs law to the same limit, i = nr, the followingexpression for b at almost normal incidence is obtained:

= ± −+ = ± −

+bi ri r

nn

( ) ( )11

. (11)

Elsewhere in the same volume of his lectures [14], Feynman makes the direct calculation ofthe field radiated along the z-axis by a plane of charges at z = 0, all coherently oscillating withfrequency ω, which we can think of as the thin glass layer of thickness Δz. Since he makesuse of the approximation − ≪( )n 1 1 to simplify the calculation, neglecting re-radiation ofre-radiation effects (multiple scattering), we can imagine the thin layer is made of a lowdensity glass that satisfies this approximation. The result for the field at a point z and time t isgiven by

Δ ωϵ= − ω −E z t i

zN qx

ce( , )

2, (12)rad

i t z c0

0

( )

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where q is the charge, x0 its maximum displacement, and N is the volume density of theoscillating charges in the layer. The layer radiates both forward and backward, andthe formula thus holds for either positive and negative z. Charge displacementfrom equilibrium gives rise to a dipole moment of modulus =p qx

0 0, and if the oscillations

in the layer are induced by an incident monochromatic plane wave, of frequency ω andunit amplitude, the material response can be described by a macroscopic polar-

ization ϵ ϵ= = − ≈ −( ) ( )P Np n n1 1 2 1 10 0 02

0 , which re-radiates a field given by

( π λ ω= =k c2 )

Δϵ Δ= − ≈ − −ω ω− −( )E z t izk

Pe i zk n e( , )2

1 (13)( ) ( )rad

i t kz i t kz

00

This is the reflected field when calculated for negative z.How does this compare with the reflected field calculated using the reflection amplitudes

b and ′b of both the front and back surfaces of the thin layer? This situation is illustrated infigure 4, and the calculation also includes the small accumulated phase delay in one round-tripof the radiation inside the layer, keeping, to the same approximation as before, only first ordercontributions.

The results is:

Δ

Δ

Δ

= + ′

≈ ± −+ − −

≈ ± − − −

≈ ± −

Δ ω

ω

ω

ω

− −

−

−

−

( )( )

( )

( )

( )

( )

E z t b e b e

nn

i kn z e

nn

i kn z e

i zk n e

( , )

( )11

1 1 2

( )1

21 1 2

( ) 1 (14)

( )

( )

( )

( )

refli kn z i t kz

i t kz

i t kz

i t kz

2

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Figure 4. Normal reflection off a thin plane-parallel glass layer.

Equations (13) and (14) agree if − is selected as the sign ±( ) of b. The final result is thus

= − −+

( )( )

bsin i r

sin i r(15)

and for B

= − −+

( )( )

Btan i r

tan i r, (16)

which embodies the existence of the external Brewsterʼs angle.The way this can be understood on the basis of our atomic hypotesis is the following. The

polarization currents of the oscillating dipoles, to which the refracted (s- and p-polarized)waves are parallel, radiate the reflected waves into the vacuum. In the case of a p-polarizedwave, the polarization current (and the associated A amplitude) oscillates in the direction ofthe reflected beam when the condition π+ =( )i r 2 is satisfied, but this is precisely thedirection along which there is no radiation emission, and so there is no reflected wave inthis case.

Confusion has arisen in the past about this interpretation, essentially because there hasbeen no clear perception about the difference between the total polarization (which includesthe dipolar interactions in the material), which radiates into vacuum, and the source polar-ization induced by the incident vacuum wave, which radiates inside the medium as a wavethat refracts at the air-glass interface. This is the how, in this case, the dipolar interaction inthe glass is accounted for. Both interpretations are correct, even if the first is the one mostcommonly used in linear optics. What is wrong is to assume that the total polarization radiatesinside the medium, since the dipolar interaction must be taken into account onlyonce [15, 16].

Expressions (15) and (16) agree with those derived directly, in more standard fashion, byMaxwellʼs equations [17], which of course take correct account of the dipolar interactions inthe material.

4. Internal Brewsterʼs angle

We have never seen Feynmanʼs derivation of b and B reproduced or cited in any textbook ofphysics or optics. This seems a little strange since we find it elegant, subtle, and useful forunderstanding, from an elementary point of view, the microscopic mechanisms that underliethe optical phenomena of reflection and refraction. Furthermore, this is the simplest way tointroduce the extinction theorem to a wider audience of students, by directly founding it onthe results of the experimental observation.

As an example of application, following the same lines of thought as developed in theprevious sections, we would like to derive and explain the internal Brewsterʼs angle, which isobtained when a light beam strikes a glass surface from inside under the same condition

π+ =( )i r 2 as for the external Brewsterʼs angle. This is fun since the reasonings areelementary, but not trivial extensions of what has been carried out before.

Figure 5 helps to investigate the case of a light beam incident on the air-glass interfacefrom inside.

If we look at figure 5(c), where the lower part has been hidden, we cannot guess wherethe incident beam is coming from on the basis of the purely geometrical features displayed,since figures 5(a) and (b) show that both initial states end up in the same final state. Indeed,

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from experiments we know that this geometrical symmetry breaks at the Brewsterʼs angle forp-polarized incident radiation since either reflection in air (in figure 5(a)) or reflection in glass(in figure 5(b)) become zero, and the upper part of figure 5(c) is no longer identical for thetwo cases. Can figure 5 be useful for guessing something about this symmetry breaking?

For the situation in figure 5(a), we have seen that the incident wave from the air induces apolarization that generates the refracted beam and, in addition, two vacuum waves (propa-gation speed c) that lie symmetrically to one another with respect to the interface. One is thereflected wave in the air, while the other propagates in the glass and extinguishes the incidentwave along its straight continuation into the glass.

With analogous reasoning, let us describe what is happening in the case depicted infigure 5(b). Since a wave excited in the glass is accompanied by two vacuum waves, we mustconcede that for the two waves of figure 5(b), the incident beam striking the glass surfacefrom inside and its reflected beam, there are four vacuum waves to deal with. Sinceexperimentally two separate vacuum waves in the air, generated by the polarization currentsassociated with the incident and the reflected beams, are not observed but just one instead, wededuce that both vacuum waves exactly overlap and propagate at an angle i to the interfacenormal to set up the refracted wave in the air.

As regards the vacuum wave in glass associated with the incident beam, it will be at anangle i (by the symmetry previously noted) to the interface normal, and will have a complextwofold role. First, it mimics the situation in figure 5(a) of a beam striking the glass surfacefrom the air, and generates the reflected field in the glass. Second, it has the role of cancellingany trace of total vacuum internal field (experimentally, none is observed), destructivelyinterfering with the internal vacuum wave of the reflected beam: again, the extinc-tion theorem.

Within this scenario, maintaining the same names for the angles, but with exchangedroles, and using ′ ′b B, and ′ ′a A, for the reflected and refracted amplitudes of s- and p-polarized light, respectively, we could be tempted, mutatis mutandis, to rewrite equation (2)for the extinction of the two vacuum waves in the glass:

′ −′ = −

−( )B cos i r

b11

(17)

where, as before, the incoming beams have unit amplitude. However, this proportion is wrongfor light incident from inside the glass.

For s-polarization, the contributions in the denominators are still ′b to the left and −1 tothe right of equation (17), respectively, since both are proportional to the incoming unitamplitude beam in glass, all the waves being parallel. Instead, in the case of p-polarized fields,

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Figure 5. Which case is depicted in (c), is it (a) or (b)?

the contributions in the numerators of equation (17) are only a fraction of ′B to the left and of−1 to the right of it, respectively, since the incoming and the reflected (as well as therefracted) fields are all at different angles with one another, as illustrated in figure 6.

The heavy dotted line represents the direction of the overlapping internal vacuum fieldsassociated with the incoming and the reflected beams. The fraction of the reflected wave ′Bcontributing to its vacuum wave is already accounted for in the left-hand side of equation(17). Through geometrical considerations in figure 6, it is seen that only the projection

+( )cos i r of the incoming wave onto the perpendicular to the propagation direction of itsassociated vacuum wave is effective in generating it to the right-hand side of equation (17).Thus, the correct expression, analogous to equation (2), is in this case

′ −′ = − +

−( ) ( )B cos i r

b

cos i r

1, (18)

and thus

′ = ′ +−

( )( )

B bcos i r

cos i r. (19)

However, ′ = −b b, and using equation (15) we obtain the final result for ′B :

′ = −+ = −( )

( )B

tan i r

tan i rB, (20)

consistent with the Stokes’ relations, and, as before, also embodying Brewsterʼs law.The usual explanation is obviously unsuitable for the internal Brewsterʼs law since there

are no oscillating dipoles in the refracting medium, which is air. A subtler analysis is neededto understand the roles of the polarization currents and their radiated waves in this case[18, 19]3,4. First, the internal Brewsterʼs angle for which ′B becomes zero is reached when

π+ =( )i r 2, exactly as in the case of the external Brewsterʼs angle. Furthermore, it is easy

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A' B'

i rr

1

glass

Figure 6. Geometrical constructions for the computation of the fractions of 1 and ′Bthat determine their internal vacuum waves for p-polarization.

3 [18] is the oldest article in English, we are aware of, in which this explanation has been presented, but it wascertainly well known in the Russian literature long since.4 In [19] the analytical path is taken, solving the integral equation of the problem.

to see that π+ =( )i r 2 also corresponds to the condition when the angle between theincident unit amplitude and the propagation direction of its associated internal vacuum waveis π 2, and so this wave is not present since no field is radiated along this direction. However,if there is no such vacuum wave, the reflected wave is not generated either, and this explainsthe internal Brewsterʼs law. Or, to put it differently, if a reflected wave could be present, itwould be accompanied by an unbalanced vacuum wave of amplitude different from zero:experimentally, neither of them are observed.

References

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complex nature of wave motion: https://www.youtube.com/watch?v=19zRwxtJSOo[5] Born M and Wolf E 1999 Principles of Optics 7th edn (Cambridge: Cambridge University Press)

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