13The Interference Laws of Fresneland Arago

13
The Interference Laws of Fresnel
and Arago

13.1

INTRODUCTION

In this last chapter of the first part, we now turn to the topic that led Stokes to
introduce his polarization parameters, namely, the mathematical formulation of
unpolarized light and its application to the interference laws of Fresnel and
Arago. In this section the events that led up to Stokes’ investigation are described.
We briefly review these events.
The investigation by Stokes that led to his paper in 1852 began with the
experiments performed by Fresnel and Arago in 1817. At the beginning of these
experiments both Fresnel and Arago held the view that light vibrations were longitudinal. However, one of the results of these experiments, namely, that two rays that
are polarized at right angles could in no way give rise to interference, greatly puzzled
Fresnel. Such a result was impossible to understand on the basis of light vibrations
that are longitudinal. Young heard of the experiments from Arago and suggested
that the results could be completely understood if the light vibrations were transverse. Fresnel immediately recognized that this condition would indeed make the
experiments intelligible. Indeed, as J. Strong has correctly pointed out, only after
these experiments had been performed was the transverse nature of light as well as
the properties of linearly, circularly, and elliptically polarized light fully understood.
The results of the Fresnel–Arago experiments have been succinctly stated as the
interference laws of Fresnel and Arago. These laws, of which there are four, can be
summarized as follows:
1.
2.
3.

Two waves linearly polarized in the same plane can interfere.

Two waves linearly polarized with perpendicular polarizations cannot
interfere.
Two waves linearly polarized with perpendicular polarizations, if derived
from perpendicular components of unpolarized light and subsequently
brought into the same plane, cannot interfere.

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4.

Two waves linearly polarized with perpendicular polarizations, if derived
from the same linearly polarized wave and subsequently brought into the
same plane, can interfere.

The fact that orthogonally polarized rays cannot be made to interfere can be
taken as a proof that light vibrations are transverse. This leads to a complete understanding of laws 1 and 2. The same confidence in understanding cannot be made with
respect to laws 3 and 4, however. For these laws involve unpolarized light, a quantity
that Fresnel and Arago were unable to understand completely or to characterize
mathematically. As a consequence, they never attempted a mathematical formulation of these laws and merely presented them as experimental facts.
Having established the basic properties of unpolarized, as well as partially
polarized light, along with their mathematical formulation, Stokes then took up
the question of the interference laws of Fresnel and Arago. The remarkable fact
now emerges that Stokes made no attempt to formulate these laws. Rather, he
analyzed a related experiment that Stokes states is due to Sir John Herschel. This
experiment is briefly discussed at the end of this chapter.
The analysis of the interference laws is easily carried to completion by means of
the Mueller matrix formalism. The lack of a matrix formalism does not preclude a
complete analysis of the experiments, but the use of matrices does make the calculations far simpler to perform. We shall first discuss the mathematical statements of
unpolarized light. With these statements we then analyze the experiments through

the use of matrices, and we present the final results in the form of the Stokes vectors.
The apparatus that was used by Fresnel and Arago is similar to that devised by
Young to demonstrate the phenomenon of interference arising from two slits. In
their experiments, however, polarizers are appropriately placed in front of the light
source and behind the slits in order to obtain various interference effects. Another
polarizer is placed behind the observation screen in two of the experiments in order
to bring the fields into the same plane of polarization. The optical configuration will
be described for each experiment as we go along.

13.2

MATHEMATICAL STATEMENTS FOR UNPOLARIZED LIGHT

In most optics texts very little attention is paid to the subject of unpolarized light.
This subject was the source of numerous investigations during the nineteenth century
and first half of the twentieth century. One of the major reasons for this interest was
that until the invention of the laser practically every known optical source emitted
only unpolarized light. Ironically, when the subject of unpolarized light was finally
‘‘understood’’ in the late 1940s and 1950s, a new optical source, the laser, was
invented and it was completely polarized! While there is a natural tendency to
think of lasers as the optical source of choice, the fact is that unpolarized light
sources continue to be widely used in optical laboratories. This observation is supported by looking into any commercial optics catalog. One quickly discovers that
manufacturers continue to develop and build many types of optical sources, including black-body sources, deuterium lamps, halogen lamps, mercury lamps, tungsten
lamps, etc., all of which are substantially unpolarized. Consequently, the subject of
unpolarized light is still of major importance not only for understanding the Fresnel–
Arago laws but because of the existence and use of these optical sources. Hence, we

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should keep in mind that the subject of unpolarized light is far from being only of
academic interest.
In all of the experiments of Fresnel and Arago an unpolarized source of light is
used. The mathematical statements that characterize unpolarized light will now be
developed, and these expressions will then be used in the analysis of the Fresnel–
Arago experiments and the formulation of their laws.
The Stokes parameters of a beam of light, as first shown by Stokes, can be
determined experimentally by allowing a beam of light to propagate through a
retarder of retardance  and then through a polarizer with its transmission axis at
an angle  from the x axis. The observed intensity I(, ) of the beam is found to be
1
Ið, Þ ¼ ½S0 þ S1 cos 2 þ S2 sin 2 cos  À S3 sin 2 sin Š
2

ð13-1Þ

where S0, S1, S2, and S3 are the Stokes parameters of the incident beam. In order to
use (13-1) to characterize unpolarized light, Stokes invoked the experimental fact
that the observed intensity of unpolarized light is unaffected by the presence of the
retarder and the orientation of the polarizer. In other words, I(, ) must be independent of  and . This condition can only be satisfied if
S1 ¼ S2 ¼ S3 ¼ 0,

S0 6¼ 0

ð13-2aÞ

so
Ið, Þ ¼ S0 =2

ð13-2bÞ

The Stokes parameters for a time-varying field with orthogonal components
Ex(t) and Ey(t) in a linear basis are defined to be
S0 ¼ hEx ðtÞExà ðtÞi þ hEy ðtÞEyà ðtÞi

ð13-3aÞ

S1 ¼ hEx ðtÞExà ðtÞi À hEy ðtÞEyà ðtÞi

ð13-3bÞ

S2 ¼ hEx ðtÞEyà ðtÞi þ hEy ðtÞExà ðtÞi

ð13-3cÞ

S3 ¼ ihEx ðtÞEyà ðtÞi À ihEy ðtÞExà ðtÞi

ð13-3dÞ

where hÁ Á Ái means a time average and an asterisk signifies the complex conjugate.
The Stokes parameters for an unpolarized beam (13-2) can be expressed in terms of
the definition of (13-3) so we have
hEx ðtÞExà ðtÞi þ hEy ðtÞEyà ðtÞi ¼ S0

ð13-4aÞ

hEx ðtÞExà ðtÞi À hEy ðtÞEyà ðtÞi ¼ 0

ð13-4bÞ

hEx ðtÞEyà ðtÞi þ hEy ðtÞExà ðtÞi ¼ 0

ð13-4cÞ

ihEx ðtÞEyà ðtÞi À ihEy ðtÞExà ðtÞi ¼ 0

ð13-4dÞ

From (13-4a) and (13-4b) we see that
1
hEx ðtÞExà ðtÞi ¼ hEy ðtÞEyà ðtÞi ¼ S0
2

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ð13-5Þ


Thus, we conclude from (13-5) that the time-averaged orthogonal quadratic field
components are equal, and so for unpolarized light we tentatively set
Ex ðtÞ ¼ Ey ðtÞ ¼ AðtÞ

ð13-6Þ

This expression indeed satisfies (13-4a) and (13-4b). However, from (13-4c) and
(13-4d) we have
hEx ðtÞEyà ðtÞi ¼ hEy ðtÞExà ðtÞi ¼ 0

ð13-7aÞ

and this cannot be satisfied by (13-6). Therefore, we must set
Ex ðtÞ ¼ Ax ðtÞ

ð13-7bÞ

Ey ðtÞ ¼ Ay ðtÞ

ð13-7cÞ

in order to satisfy (13-4a) through (13-4d). We see that unpolarized light can be
represented by
hAx ðtÞAÃx ðtÞi ¼ hAy ðtÞAÃy ðtÞi ¼ hAðtÞAÃ ðtÞi

ð13-8aÞ

hAx ðtÞAÃy ðtÞi ¼ hAy ðtÞAÃx ðtÞi ¼ 0

ð13-8bÞ

and

Equations (13-8) are the classical mathematical statements for unpolarized
light. The condition (13-8b) is a statement that the orthogonal components of unpolarized light have no permanent phase relation. In the language of statistical analysis,
(13-8b) states that the orthogonal field components of unpolarized light are uncorrelated. We can express (13-8a) and (13-8b) as a single statement by writing
hAi ðtÞAÃj ðtÞi ¼ hAðtÞAÃ ðtÞi Á ij

i, j ¼ x, y

ð13-9aÞ

where ij is the Kronecker delta defined by

13.3

ij ¼ 1

if i ¼ j

ð13-9bÞ

ij ¼ 0

if i 6¼ j

ð13-9cÞ

YOUNG’S INTERFERENCE EXPERIMENT WITH
UNPOLARIZED LIGHT

Before we treat the Fresnel–Arago experiments, we consider Young’s interference
experiment with an unpolarized light source using the results of the previous section.
In many treatments of Young’s interference experiments, a discussion of the nature
of the light source is avoided. In fact, nearly all descriptions of the experiment in
many textbooks begin with the fields at each of the slits and then proceed to show
that interference occurs because of the differences in path lengths between the slits
and the screen. It is fortuitous, however, that regardless of the nature of the light
source and its state of polarization, interference will always be observed. It was
fortunate for the science of optics that the phenomenon of interference could be
described without having to understand the nature of the optical source. Had optical
physicists been forced to attack the problem of the polarization of sources before

proceeding, the difficulties might have been insurmountable and, possibly, greatly

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


impeded further progress. Fortunately, this did not occur. Nevertheless, the problem
of characterizing the polarization of light remained a problem well into the twentieth
century as a reading of the papers in the references at the end of this chapter show.
Many beginning students of physical optics sometimes believe that Young’s
experiment must be performed with light that is specially prepared; i.e., initially the
light source is unpolarized and then is transformed to linear polarized light before it
arrives at the slits. The fact is, however, that interference phenomena can be
observed with unpolarized light. This can be easily shown with the mathematical
statements derived in the previous section.
In Young’s experiment an unpolarized light source is symmetrically placed
between the slits A and B as shown in Fig. 13-1. The Stokes vector of the unpolarized
light can again be decomposed in the following manner:
0 1
0 1
0
1
1
1
1
B0C 1
B C 1
B
C
à B1C
à B À1 C

C
S ¼ hAAÃ iB
ð13-10Þ
@ 0 A ¼ 2 hAx Ax i@ 0 A þ 2 hAy Ay i@ 0 A
0
0
0
The Stokes vector at slit A is
0 1
0
1
1
1
B1C 1
B
C
1
à B À1 C
C
SA ¼ hAx AÃx iB
@ 0 A þ 2 hAy Ay i@ 0 A
2
0 A
0 A

ð13-11aÞ

and at slit B is
0 1
0

1
1
1
B1C 1
B
C
1
C þ hAy AÃy iB À1 C
SB ¼ hAx AÃx iB
@
@
A
A
0
0
2
2
0 B
0 B

Figure 13-1 Young’s interference experiment with unpolarized light.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð13-11bÞ


where the subscripts A and B remind us that these are the Stokes vectors of the field
at the respective slits.
The fields which satisfy the Stokes vector SA are

Ax ðtÞ
ffiffiffi
ExA ðtÞ ¼ p
2

Ay ðtÞ
EyA ðtÞ ¼ pffiffiffi
2

ð13-12aÞ

Ay ðtÞ
EyB ðtÞ ¼ pffiffiffi
2

ð13-12bÞ

and SB
Ax ðtÞ
ffiffiffi
ExB ðtÞ ¼ p
2

The field components at point C on the screen arising from the field propagating
from slit A is
Ax ðtÞ
ffiffiffi expðiA Þ
ExA ðtÞ ¼ p
2

ð13-13aÞ

Ay ðtÞ
EyA ðtÞ ¼ pffiffiffi expðiA Þ
2

ð13-13bÞ

and, similarly, that due to slit B
Ax ðtÞ
ffiffiffi expðiB Þ
ExB ðtÞ ¼ p
2

ð13-14aÞ

Ay ðtÞ
EyB ðtÞ ¼ pffiffiffi expðiB Þ
2

ð13-14bÞ

The total field in the x and y directions is
Ax ðtÞ
ffiffiffi ½expðiA Þ þ expðiB ފ
Ex ðtÞ ¼ ExA ðtÞ þ ExB ðtÞ ¼ p
2

ð13-15aÞ

Ay ðtÞ
Ey ðtÞ ¼ EyA ðtÞ þ EyB ðtÞ ¼ pffiffiffi ½expðiA Þ þ expðiB ފ
2

ð13-15bÞ

Ax ðtÞ
ffiffiffi ð1 þ ei Þ
Ex ðtÞ ¼ p
2

ð13-16aÞ

Ay ðtÞ
Ey ðtÞ ¼ pffiffiffi ð1 þ ei Þ
2

ð13-16bÞ

and

or

where  ¼ B À A and the constant factor expðiA Þ has been dropped. Equation
(13-16) describes the field components at a point C on the observing screen. It is
interesting to note that it is not necessary at this point to know the relation between
the slit separation and the distance between the slits and the observing screen. Later,
this relation will have to be known to obtain a quantitative description of the interference phenomenon. We shall see shortly that interference is predicted with the
information presented above.

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The Stokes vector for (13-16) is now formed in accordance with (13-3) and
applying the conditions for unpolarized light (13-8) or (13-9). We then find that the
Stokes vector for the field at C is
0 1
1
B
0C
C
ð13-17Þ
S ¼ hAAÃ ið1 þ cos ÞB
@0A
0
Thus, we see from (13-17) that light observed on the screen is still unpolarized.
Furthermore, the intensity is
I ¼ hAAÃ ið1 þ cos Þ

ð13-18Þ

Equation (13-18) is the familiar statement for describing interference. According to
(13-18), the interference pattern on the screen will consist of bright and dark (null
intensity) lines.
In order to use (13-18) for a quantitative measurement, the specific relation
between the slit separation and the distance from the slits to the screen must be
known. This is described by  ¼ B À A ¼ kÁl, where k ¼ 2/ and Ál is the path
difference between the fields propagating from A and B to C. The phase shift can be
expressed in terms of the parameters shown in Fig. 13-1.


a 2
l22 ¼ d 2 þ y þ
ð13-19aÞ
2

a 2
l12 ¼ d 2 þ y À
ð13-19bÞ
2
Subtracting (13-19b) from (13-19a) yields
l22 À l12 ¼ 2ay

ð13-20Þ

We can assume that a is small, d ) a, and c is not far from the origin so that
l2 þ l1 ffi 2d

ð13-21Þ

so (13-20) becomes
Ál ¼ l2 À l1 ¼

ay
d

ð13-22Þ

The phase shift  is then
 ¼ B À A ¼ kÁl ¼

2ay
d

ð13-23Þ

where k ¼ 2/ is the wavenumber and  is the wavelength of the optical field. The
maximum intensities are, of course, observed when cos  ¼ 1,  ¼ 2m so that
 
d
y¼
m m ¼ 0, 1, 2, :::
ð13-24aÞ
a

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and the minimum (null) intensities are observed when cos  ¼ À1,  ¼ ð2m þ 1Þ
so that
 
d
1 3 5
ð13-24bÞ
y¼
m
m ¼ , , , :::
a
2 2 2
One can easily show that, regardless of the state of polarization of the incident
beam, interference will be observed. Historically, this was first done by Young and

then by Fresnel and Arago, using unpolarized light.
We now consider the mathematical formulation of the Fresnel–Arago interference laws.

13.4

THE FIRST EXPERIMENT: FIRST AND SECOND
INTERFERENCE LAWS

We consider a source of unpolarized light  symmetrically placed between slits A and
B as shown in Fig. 13-2. A linear polarizer P with its transmission axis parallel to
the x axis is placed in front of the light source. A pair of similar polarizers PA and PB
are also placed behind slits A and B, respectively. The transmission axes of these
polarizers PA and PB are at angles  and
with respect to the x axis, respectively. We
wish to determine the intensity and polarization of the light on the screen Æ.
The Stokes vector for unpolarized light of intensity AA* can be represented by
0 1
1
B
0C
C
S ¼ hAðtÞAÃ ðtÞiB
ð13-25Þ
@0A
0

Figure 13.2 The first experiment. The transmission axis of the P is parallel to the x axis.
The transmission axes of PA and PB are at angles  and
from the x axis.

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Equation (13-25) can be decomposed into two orthogonally linearly polarized
beams. We then write
0 1
1
B0C
B C
S ¼ hAAÃ iB C
@0A
0
0 1
0
1
1
1
B1C 1
B À1 C
1
B C
B
C
¼ hAx AÃx iB C þ hAy AÃy iB
C
@
A
@
2
2

0
0 A
0

ð13-26Þ

0

where we have used (13-8a).
The Mueller matrix for P is
0
1
1 1 0 0
1B1 1 0 0C
C
M ¼ B
2@0 0 0 0A
0 0 0 0

ð13-27Þ

The output beam from P is obtained from the multiplication of (13-27) and (13-26):
0 1
1
B
1
1C
C
SP ¼ hAx AÃx iB
ð13-28Þ

@0A
2
0
Thus, the polarizer P transmits the horizontal and rejects the vertical component of
the unpolarized light, (13-26). The light is now linearly horizontally polarized.
The matrix of a polarizer, MP, with its transmission axis at an angle  from the
x axis, is determined from
MP ð2Þ ¼ MðÀ2ÞMP Mð2Þ

ð13-29Þ

where MP(2) is the matrix of the rotated polarizer and M(2) is the rotation matrix:
0
1
1
0
0
0
B 0 cos 2 sin 2 0 C
B
C
Mð2Þ ¼ B
ð13-30Þ
C
@ 0 À sin 2 cos 2 0 A
0
0
0
1
The Mueller matrix for PA is then found by setting  ¼  in (13-30) and then

substituting (13-27) into (13-29). The result is
0
1
1
cos 2
sin 2
0
cos 2 2
cos 2 sin 2 0 C
1B
B cos 2
C
MP A ¼ B
ð13-31Þ
C
2 @ sin 2 cos 2 sin 2
0A
sin 2 2
0
0
0
0

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A similar result holds for MPB with  replaced by
. The Stokes vector SA that
emerges from PA is obtained by the multiplication of (13-28) by (13-31):
0

1
1
B cos 2 C
1
B
C
SA ¼ hAx AÃx i cos 2 B
ð13-32aÞ
C
@ sin 2 A
2
0
In a similar manner the Stokes vector SB is found to be
0
1
1
B cos 2
C
1
B
C
SB ¼ hAx AÃx i cos 2
B
C
@ sin 2
A
2
0

ð13-32bÞ

Inspection of (13-32a) and (13-32b) shows that both beams are linearly polarized at
slits A and B.
In order to describe interference phenomena at the screen Æ, we must now
determine the fields at slits A and B in the following manner. From the definition of
the Stokes vector given by (13-3) and the Stokes vector that we have just found at slit
A, Eq. (13-32a), we can write
1
hEx ðtÞExà ðtÞiA þ hEy ðtÞEyà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2 
2
1
hEx ðtÞExà ðtÞiA À hEy ðtÞEyà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2  cos 2
2
1
hEx ðtÞEyà ðtÞiA þ hEy ðtÞExà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2  sin 2
2

ð13-33bÞ

ihEx ðtÞEyà ðtÞiA À ihEy ðtÞExà ðtÞiA ¼ 0

ð13-33dÞ

ð13-33aÞ

ð13-33cÞ

where the subscript A on the angle brackets reminds us that we are at slit A. We now
solve these equations and find that
1

hEx ðtÞExà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos4 
2
1
hEy ðtÞEyà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2  sin 2 
2
1
hEx ðtÞEyà ðtÞiA ¼ hEy ðtÞExà ðtÞiA ¼ hAx ðtÞAÃx ðtÞi cos 2  sin 2
4

ð13-34aÞ
ð13-34bÞ
ð13-34cÞ

We see that the following fields will then satisfy (13-34):
Ax ðtÞ
ffiffiffi cos 2 
ExA ðtÞ ¼ p
2

ð13-35aÞ

Ax ðtÞ
ffiffiffi cos  sin 
EyA ðtÞ ¼ p
2

ð13-35bÞ

where Ax(t) is the time-varying amplitude. The quantity Ax(t) is assumed to vary
slowly in time. In view of the fact that the Stokes vector at slit B is identical in form

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


with that at slit A, the field at slit B, following (13-35), will be
Ax ðtÞ
ffiffiffi cos 2

ExB ðtÞ ¼ p
2

ð13-36aÞ

Ax ðtÞ
ffiffiffi cos
sin

EyB ðtÞ ¼ p
2

ð13-36bÞ

The propagation of the beams along the paths AC and BC as shown in Fig. 13-2
increases the phase of the fields by an amount A ¼ kl1 and B ¼ kl2 , respectively,
where k ¼ 2/ and  is the wavelength. Thus, at point C on the screen Æ, the s and p
field components will be, by the principle of superposition,
Ex ðtÞ ¼ ExA ðtÞ expðiA Þ þ ExB ðtÞ expðiB Þ

ð13-37aÞ

Ey ðtÞ ¼ EyA ðtÞ expðiA Þ þ EyB ðtÞ expðiB Þ

ð13-37bÞ

Ex ðtÞ ¼ expðiA Þ½ExA ðtÞ þ ei ExB ðtފ

ð13-38aÞ

or

i

Ey ðtÞ ¼ expðiA Þ½EyA ðtÞ þ e EyB ðtފ

ð13-38bÞ

where  ¼ B À A ¼ kðl2 À l1 Þ: The factor expðiA Þ will disappear when the Stokes
parameters are formed, and so it can be dropped. We now substitute (13-35) and
(13-36) into (13-38), and we find that
Ax ðtÞ
ffiffiffi ðcos 2  þ ei cos 2
Þ
Ex ðtÞ ¼ p
2

ð13-39aÞ

Ax ðtÞ
ffiffiffi ðcos  sin  þ ei cos
sin

Þ
Ey ðtÞ ¼ p
2

ð13-39bÞ

The Stokes parameters for Ex ðtÞ and Ey ðtÞ are now formed in the same manner as in
(13-3). The Stokes vector observed on the screen will then be
0
1
cos 2  þ cos 2
þ 2 cosð À
Þ cos  cos