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2 Light as a Wave

# 2 Light as a Wave

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What Is Light?

a

b

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c

Fig. 1.6 Constructive and destructive interference. (Left)

Pairs of overlapping waves. (Right) The sum of the two

components. (a) Constructive interference with the same

phase with the maximum sum. (b) 150° phase shift; the

resulting amplitude is weaker. (c) Cancellation with a

phase difference of 180° (half wavelength) in that the

peaks coincide with the valleys

Fig. 1.7 A wave, coming from the left, passes through a

single, small opening and spreads out to the right due to

diffraction. Averaged over a few vibrational periods, the

screen (at the right) becomes uniformly illuminated. I

intensity on the screen

the wave properties of light are somewhat hidden

from the eye due to the very small wavelengths

involved (0.4–0.8 mm). The wave properties of

light are proven most convincingly, though, with a

phenomenon that is observable only with waves:

light can cancel out light, just as a peak and a valley of two overlapping water waves neutralize one

another. This phenomenon is called interference

(Fig. 1.6) and forms the basis for the double slit

experiment that we shall now discuss.

1.2.1

The Double Slit Experiment

Young’s double slit experiment is deemed decisive evidence of the wave nature of light. Let us

start with a single slit. Due to diffraction, as light

passes through a single tiny opening, it fans out

and produces a uniform illumination of the screen

(Fig. 1.7). If a second nearby hole or slit is

opened, a pattern of stripes appears on the screen

(Fig. 1.8). At certain places, the light coming

from the two openings is extinguished. These are

precisely the locations where the path differences

from the two openings amounts to half a wavelength: a wave crest meets a wave depression.5

l

Fig. 1.8 The double slit experiment. Light coming from

the left. The screen at the right is illuminated by the light

coming from the two openings. An interference pattern

appears on the screen. Cancellation (destructive interference) occurs at those locations where the paths from the

two openings differ by an odd number (1, 3, 5, …) of half

wavelengths. The lengths of the red paths differ by 5/2

wavelengths. I Intensity on the screen

From the geometric arrangement and the interference pattern, Young could even derive the

wavelength. He found the values given above.

But what is it that actually vibrates? That was the

big question of his time.

1.2.2

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Young did not carry out his experiment with two openings

but split a sun ray with a piece of paper.

A Freehand Interference

Experiment

The principle of the interference experiment with

light can be observed with the simplest means – with

1.2

Light as a Wave

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the naked eye (Fig. 1.9). The invested time will be

repaid by acquiring a more direct relationship with

the wave nature of light. Poke two tiny holes into a

piece of paper, as close together as possible and, at

night, observe a small, bright source of light (e.g., a

street lamp at a large distance) through these openings. Through the interference of the light coming

from the two openings, a striped pattern arises on

the retina that appears to be about the size of the

moon. The dark areas arise there where the difference in pathway from the two openings amounts to

an odd number of half wavelengths – light waves

with a phase difference of 180° extinguish each

other. In terms of electrodynamics, at this point,

electrical fields with opposite directions meet. The

clearest patterns can be seen with monochromatic

light. As light sources, the yellow-orange street

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Fig. 1.9 An easy version of Young’s double slit

experiment: observing light interference with the simplest

means. (1) Light from a distant street lamp. (2) Apertures,

consisting of two needle holes (ca. 0.2 mm diameter) close

together (ca. 0.6 mm) in a piece of paper. (3) Interference

pattern on the retina (ca. 0.2 mm diameter, corresponding

to 0.5°)

lamps (sodium vapor lamps, l = 588 nm), for

instance, are very well suited. The mechanism is

exactly the same as in Young’s double slit experiment. For a distance between the holes of 6 mm, the

bright stripes on the retina have a separation of

roughly 10 mm, corresponding to a visual angle of

about 2 min of arc. A model for this demonstration

is an apparatus, invented in 1935 by Yves Le Grand,

for the interferometric determination of visual acuity (see Sect. 4.2).

1.2.3

Diffraction

A further manifestation of the wave nature of

light is diffraction: When encountering an edge

or passing through an aperture, the light’s pathway is bent, i.e., deviated from its normal straight

line of travel. For example, the play of colors

seen when looking at a distant street lamp through

an opened umbrella is due to diffraction as the

light passes through the periodic arrangement of

the fibers of the umbrella textile. Diffraction also

occurs when light passes through the pupils of

our eyes. This results in a fundamental limitation

of visual acuity with small pupils (Sect. 19.1).

Due to diffraction, the resolution of a light microscope is also restricted to structures the size of a

half wavelength (Sect. 19.4).

Diffraction occurs with every wave phenomenon. With surface waves on water, they can be

observed directly: for example, when water waves

pass through an opening (Fig. 1.10). For openings

Fig. 1.10 Diffraction of water waves when passing through a harbor entrance. The larger the entrance is in comparison

with the wavelength, the less apparent the diffraction will be

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Fig. 1.11 Diffraction rings (3) on the retina created by

light (1) coming from a point-shaped source and through

a tiny hole (2). As an aperture, a tiny hole is made by

sticking a needle point through a piece of paper and this is

then held very near the eye

that are much larger than the wavelength, the

wave continues without the diffraction being

noticeable. However, the narrower the opening

is in comparison to the wavelength, the more

pronounced the deviation of the wave’s direction

will be. In the limiting case of an arbitrarily small

opening, the wave on the other side spreads out

with the same intensity in all directions.

As a variation of the experiment in Sect. 1.2.2,

we can try seeing the diffraction image made by

a round aperture. This is done by viewing a distant light source (approximating a point source)

through a tiny hole (Fig. 1.11).

1.3

What Is Light?

fields affect magnetized needles and electric

fields exert force on electrically charged particles,

e.g., on free electrons or ions. There is an electric

field, for example, between the two poles of an

electric plug. If the two poles come close enough,

the electric field between them is so strong that

sparks will be produced in the gap.

A bar magnet produces a magnetic field. It can

be perceived by a magnetized needle (such as in a

compass) that aligns itself with the direction of

the magnetic field. We now consider the situation

in which a bar magnet rotates. It generates a magnetic field such that its direction and strength will

change at every fixed location. It will oscillate in

step with the rotation. Now the laws of electrodynamics take effect: a changing magnetic

field engenders an electric field. The rotating

magnet also creates an electric field that again

oscillates in step with the rotation. Now, another

of the basic laws of electrodynamics enters: for its

part, a changing electric field once again creates a

magnetic field. This mutual creation of changing

fields propagates in space with the speed of light.

Light as an Electromagnetic

Phenomenon

Interference and the diffraction of light can be

explained by assuming that light is a wave phenomenon without being specific about the precise

nature of the vibrations that propagate through

empty space in the form of light. We will start with

a clear proposition: light is an electromagnetic

wave. The waves that travel back and forth between

mobile telephone transmitters and cell phones are

also electromagnetic waves – the difference lies

only in the wavelengths: the physical laws behind

them are exactly the same (Fig. 1.12).

As indicated by the section title, a light ray can

be understood as a combination of very rapidly

oscillating electric and magnetic fields that propagate in empty space at the speed of light. How

can we imagine these fields? Briefly put, magnetic

λ = (0.4 – 0.7) μm

λ = 33 cm

Fig. 1.12 Light as an electromagnetic wave. The difference

between light and cell phone waves lies in their respective

wavelengths l (cell phone: l = 33 cm, light: l = 0.4–0.7 mm)

1.3

Light as an Electromagnetic Phenomenon

We will not consider the process of the propagation in more detail, but sound and water waves

also propagate away from a local disturbance.

With the rotation of the bar magnet, we create

an electromagnetic wave that spreads out in all

directions at the speed of light. At every fixed

location in space, it vibrates in step with the rotation. If the bar magnet were to be rotated with a

frequency of 108 Hz, radio waves would be produced – and when it is rotated even more rapidly,

with a rotation frequency of 5·1014 Hz, yellow

light would be seen. Only atoms, though, can

achieve such frequencies.

Initially, Maxwell’s6 1864 hypothesis that

light consists of electromagnetic waves was

purely speculative: at the time, electromagnetic

waves were not known but only a possible solution to his equations, resulting from his mathematics. We pay tribute to this event here by

concerning ourselves with it a bit further. The

empirical foundation was created by the great

experimenter Faraday7 in the first half of the

nineteenth century with his research regarding

the emergence of electric and magnetic fields

from electric charges and currents, as well as the

discovery of the laws of induction (changing

magnetic fields create electric fields – the basis

for transformers). Maxwell succeeded in comprehending all of these phenomena quantitatively

with his four equations.8 In addition, far beyond

the laboratory experiments, they exhibited –

purely mathematically – a noteworthy solution:

electromagnetic waves of any desired wavelength

that propagate in a vacuum with a speed of

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James Maxwell (1831–1879), Scottish physicist and

mathematician. Creator of the fundamental equations of

electrodynamics that are still exactly the same today. In a

lecture, with three projectors, he demonstrated additive

color mixing.

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Michael Faraday (1791–1867), English chemist and

physicist, investigator of the fundamentals of electricity

and electromagnetic fields.

8

His laws convey, in mathematically exact form, the fact

that electric charges and changing magnetic fields create

electric fields – electric currents and changing electric

fields are sources of magnetic fields. Maxwell’s equations

are still valid today and are unchanged; they have even

survived the “storm” of the special theory of relativity.

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Fig. 1.13 Michael Faraday

Fig. 1.14 James C. Maxwell

ca. 300,000 km/s – if they were to exist. This

speed resulted from the constants measured in

the laboratory concerning the relationships

between charges, currents, and fields. The agreement with the known velocity of light was

spectacular (Figs. 1.13 and 1.14).

In Fig. 1.15, we illustrate an electromagnetic

light wave. This is a snapshot of the wave for a

moment in time. The whole aggregate would be

moving with the velocity of light. This is an especially simple example of a light wave. It has a

specific wavelength and does not consist of various

colors, and the electric field always vibrates in the

same direction. The same is true of the magnetic

field. For this reason, we say that such a special

wave, like that shown in Fig. 1.15, is linearly polarized. We call the plane in which the magnetic field

vibrates the plane of polarization. We shall take up

other polarizations in Sect. 1.6. We should not

imagine a sunbeam as being so simple; however, it

consists of a chaotic overlapping of such waves

with all the various wavelengths in the visual range

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2 Light as a Wave

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