Hitherto, we have only spoken of the ray of light, its direction and its change of direction by reflection and diffraction. However, the ray - a straight line - which only receives physical meaning by the wave surface, is only a geometrical concept. Hence the section of Optics treated so far is called Geometrical Optics. Its results rest on the assumption that light propagates along straight lines, is reflected and refracted according to definite laws. A hypothesis concerning the mechanism of light must therefore be compatible with these laws of propagation, reflection and refraction. It is the wave theory: The laws for refraction and reflection can be explained on the basis of Huygens' principle, the straight line propagation by the smallness of the wave lengths. It also explains - the corpuscular theory cannot do this! - the interference of light without difficulty and it is just this which has made it for one century the sole theory of Physical Optics. However, the last 30 years (in 1935!) have encountered phenomena, which it cannot explain, but the corpuscular theory (Newton's emission theory) can explain. The union of these two theories is the principal task of physics.
Interference phenomena and diffraction of light
The concept that light is a wave motion is justified in that you can according to a plan produce with light a process, as Fig. 297 describes for water waves and as we know already from sound. However, you cannot achieve this with two rows of arbitrarily illuminated points. In fact, you must at a point, which is able to oscillate and which is to stay at rest, induce the rest at each instant in such a manner, that the two inducements, which the waves apply to the point, are at every instant equally large and opposite (at those points of the interference region at which enduringly twice as large a lowering and raising is to occur as if the point were only affected by one of the waves, the driving forces must be equally large and have the same direction). In other words: In order to yield interference phenomena, the light wave systems must be continuously interlinked in such a manner, that at each instant the state of oscillation of the one is in complete agreement with that of the other, especially so, that a change of any kind, which occurs in the one system, occurs at the same instant in the same manner in the other system (coherence of wave systems). You can never achieve this agreement between waves of light, which come from two independent of each other sources of light*.
*Its presence during corresponding water wave events is linked to the length of the waves.
Light wave coherence and
interference in Fresnel's mirror experiment
You can generate coherent light waves as follows (Fig. 742): AS and AS' are two mirrors, which meet at a very obtuse (almost straight) angle, L is a bright, straight source of light (a slit). Let the light source be mono-chromatic, for example, yellow. The light waves from L, which meet the mirror, are reflected as if they were coming from the mirror images of L: Those reflected by AS, as if they were coming from L', those reflected by AS', as if they were coming from L". L' and L" are thereby coherent sources of light; every contingent change in the one occurs at the same instant also in the other, since both depend in the same manner on the source of light L. The wave systems around L' and L" interlace in the angular space O - the region of interference. Place a screen** a few metres from the mirror, parallel to the common edge of the mirrors and at equal distances from the two mirror images. You see then on the section nm of the screen in the interference region - this is the purpose of the experiment! - a sequence of narrow vertical bands - bright and dark ones alternatingly - symmetrical to both sides of a bright band OO, centred between n and m. (You can bring the retina of your eye to the position of the screen and look at the bands through a magnifying glass.) The bands are both bounded clearly, the bright ones are brightest, the dark ones darkest at their centres and from the brightest place of the one band occurs a gentle grading to the darkest place of the next band. If you bring the screen closer to the mirrors, the bands approach each other. If you let the solar spectrum pass through a fine slit which serves at L as source of light, so that the slit glows sequentially in all colours, the bands are broadest as the colour of the slit is red, but become for each subsequent colour narrower; they are smallest for violet.
**In the figure, the screen has been flipped into the horizontal plane, in order to display the bands on it.
These are the remarkable facts which were yielded by the experiment of Fresnel in 1824. Their interpretation is assisted by the interference of two wave systems, which spread on the same water surface (Fig. 297), and they can be described in almost the same manner, if we stick also here to what happens in a horizontal plane (corresponding to the water surface), as Fig. 742 shows. The line NM, in which the horizontal plane intersects the screen, contains (Fig. 743) a number of short horizontal lines, intermittently dark and bright, symmetrical with respect to a bright one at the centre of n and m. The lines fuse into each other, corresponding to the bands described above. If the screen is brought closer to the mirrors, they come closer to each other, and vice versa. If in Fig. 744 L' and L" denote the mirror images, MN the screen, a···b the centres of those lines of light, you discover: If the screen is moved from MN to M'N', the centers of those lines move from the position ab to a"···b" along aa', bb'. etc. on hyperbolae with the focal points L' and L". This demonstrates the similarity of this phenomenon with the earlier ones and characterizes them as wave interference.
You see the same curves in Fig. 297, firstly, as those lines which all along contain the points at rest, and secondly as those lines which all along contain the points of strongest motion. The points L' and L" in Fig. 742 correspond to the centres A and B of the wave systems. The curves lie in pairs (11 22 33 ···) symmetrically with respect to a straight line OO at the middle between the centres of excitation. The curves 1,3 ··· in Fig. 742 correspond to the curves with the points at rest of the water surface, the curves 0, 2, ··· to the curves with the points in strongest motion. It is characteristic for every curve that everyone of their points is differently far away from L' and L", but the difference of these distances is for every point on the same curve equal; it is for each point of the curve
| 1 | 3 | 5 | 2 | 4 | 6 | |||||||
| 1/2·l | 3/2·l | 5/2·l | 2/2·l | 4/2·l | 6/2·l |
The hyperbola (Fig. 745) is the plane curve the points of which have this property (it is a conic section like the parabola and ellipse); it is symmetrical with two branches which go to infinity. The points F and F' are their focal points; they correspond to the points L' and L" in Fig. 744. The curve is characterized by the fact that
P1F - P1F' = P2F - P2F' = P3F - P3F' = · · · = const,
where P denotes points on the curve.
We return to Fig. 744. The curves are hyperbolae with the focal points L' and L". Everyone of the points of the same curve lies therefore at the same, but different distance from L' and L". The points of the line OO are at the same distance from from L' and L", that is, the difference of their distance from L' and L" is zero. The point on the screen through which passes OO, is bright, equally so on both sides the points 2., 4., 6. In contrast, the points 1., 3., 5. are dark. The wave theory of light views this regular sequence of bright and dark spots on the screen to be a counterpiece to the regular sequence of sites of strongest motion and of sites of perfect rest on the water surface (Fig. 297) and interprets it as follows: A system of ether waves spreads out from L' and L" as excitation centres (like the water waves around A and B); while we do not see the waves of the ether, we do see the result of their action - as light. We do not sense he billion times per second ups and downs of the ether waves. (Our eye cannot follow these changes, it receives from a source of light a uniform impression.) However, if there should spread around two separate points two coherent systems of ether waves, which affect each other, then they must reinforce and weaken each other. Wherever they meet at equal phase, that is, they give lastingly the impression of increased brightness; wherever they meet at opposite phase, they must weaken each other, that is, create the impression of decreased brightness. Increased (decreased) brightness takes here the place of stronger (weaker) motion.
The points with the strongest motion, that is, also of the largest
brightness (o), are those, the distances of which from L'
and L" differ by 0, 1, 2,
3 ··· complete
wave lengths l, - and those without motion,
that is, of total darkness (
), the distances of which from L' and L"
differ by 1, 3, 5 ··· half wave lengths l. At the points in
between two
neighbouring o and , the waves meet at such a phase that they
provoke between o and any brightness. For
example, between the first o and the first lie all points, the
distances of which from L' and L" differ
by more than zero and less than 1/2·l. Between the first and the second o lie all points,
the distances of which from L' and L" differ
by more than 1/2·l and
less than 1·l, etc.
At the end, we have only spoken of points on the screen, previously, however, of bands, because we have only spoken of phenomena in a horizontal plane, which was to be viewed as cut through the experimental set-up (Fig. 743) and through the screen. However, what applies to this horizontal plane, applies to every neighbouring plane above or below it. The points of the screen, of which we spoke in the end, are in every horizontal plane through the entire set-up at the same spot, whence they form together vertical lines on the screen, which represent the earlier described bands.
Fresnel's experiment tells us:
1. There exist waves of light,
2. the waves have very different lengths, and
3. the wave length
differs depending on the colour of
the light, or, conversely: The colour of
the light depends on the length of the wave. The numerical
relationship between the wave length l and the band width b is
given by the (here sufficiently exact) formula (which we quote
without proof:
l = b ·a/d or b = d/a·l,
where a is the distance from each other of the mirror images L' and L", and d the distance of the screen from the plane of the mirror images. If you know a, d and b , you can compute from this formula the wave length (but, as regards accuracy, this approach is greatly surpassed by others). In red, it is about 0.000,760 mm, in violet, about 0.000,390 mm, whence you must make a very small compared with d, in order to generate the bands. If you use a = 1 mm and d = 5 m, b = 5000·l, that is, for red about 3.8 mm, for violet, about 2 mm.
If you illuminate the slit with white light, that is, with all colours of the spectrum simultaneously, then systems of bands of different width and colour are superimposed on the screen. Its centre has a white band with coloured edges, because then bright bands of all colours are superimposed. However, the bands become narrower from red to violet; starting from the centre of the red band towards the edge less and less colours are therefore superimposed, so that the colour differs more and more from white and is red at the edge.
Fresnel's bi-prisms (Fig. 746) and Billet's half-prisms (Fig. 747) split the source of light L by refraction into two coherent source of light L'
and L" and generate for them a region of
interference (shaded in the figure).They are now only significant
for historic and instructive purposes; in contrast, highly
practical is the set-up of Michelson
of a vertically placed, plane parallel glass plate
in between two, at right angle to each other, vertical mirrors,
to which the plate forms an angle of 45º (Fig.
748). It is the beginning of his interferometer
(1880), a very important physical measuring device.
The light coming from S meets the (slightly
silver-plated) surface of the glass plate A. The plate
lets half of it pass and reflects the other half and thus splits
the bundle B into two coherent bundles 1 and 2;
it lets 1 pass and reflects 2. 1 meets
perpendicularly the mirror I, 2 perpendicularly the
mirror II. Each is reflected into itself and returns to
the plate, which lets 1 partly pass towards the source
of light and reflects 2 partly there. The remaining
bundles 1' and 2' of the bundles 1 and
2 interfere with each other; you inspect the
interference figures with a telescope. If I and II
were exactly perpendicular to each other, that is exactly perpendicular to the
bundles 1 and 2, they would simply conjugate,
but never yield interference. They interfere, because the mirrors
are not exactly at a right
angle to each other: In the telescope, you see I reflected
in A and I' as this mirror image. There always exists a deviation, which
causes II and the mirror image I' to coincide,
but to form a wedge with an extremely small angle (of the order
of 1" or even less). In this way, the bundles I and
II cause the bundles 1' and 2' to interfere. The distance
between the interference bands is b = l/2j,
where j is the angle of the
wedge referred to above and l is
the wave length.
Every transparent substance is coloured in reflected daylight. whenever it forms a sufficiently thin layer, for example, a soap bubble, a drop of oil, which spreads out on water, glass which has been blown into a very thin skin. This phenomenon is referred to as colours of thin plates; it arises by interference: Let CD in Fig. 749 be a very thin, transparent plate, its faces 1 and 2 be parallel to each other (the phenomenon also depends on the mutual inclination of the faces of the plate). Let the plates be illuminated by homogeneous light, say, purely red light by bundles of parallel rays. The ray A - representative of the bundle - is reflected by the face 1 as ray A1 - but only partially; its other part penetrates into the plate, is refracted, reflected by the face 2 and eventually emerges parallel to A1 from the plate as ray A2 (this applies to every ray parallel to A). In order to explain this process clearly, the figure shows a thick plate; however, in fact, we are only concerned with thicknesses of a few ten thousandths of a millimetre. A1 and A2 then almost coincide. As in Fresnel's experiment, they have arisen from the same ray A, whence they are capable of interference. If A1 and A2 meet an eye, accommodated to parallel rays, that is, if the eye views in the opposite direction of the interfering rays the location of the plate, from which they emerge, it sees the location somewhat lighted up, the brightness depending on how the two systems of waves encounter each other, that is, whether they make it into a locus of strongest motion or of total rest or intermediate motion. If they annul each other, no light comes to the eye from this location and the plate appears to be dark. If they do not annul each other, the local brightness lies between total darkness and maximum brightness. What happens depends on the following: The wave exiting at c only joins the at a reflected wave after it has covered the path abc. Whether it has at the end of this path the same phase as the one from a or not depends on the length of the path, that is, on the thickness of the plate and the light's angle of incidence. Let the two waves have opposite phases, that is, the rays A1 and A2 annul each other. What applies to A is also valid for every ray parallel to A. Hence, if the plate, as has been assumed, has parallel faces, every ray, which enters the plate parallel to A, covers inside the plate the same distance; every ray is split into two, which annul each other, and the entire plate appears to the unaccommodated eye to be completely dark. Note: Only when the light enters in this direction. If the direction is changed, also the length of the path inside the plate changes, and the rays need no longer annul each other. A transparent, thin plate with parallel faces which you turn in homogeneous light to and fro, reflects therefore (depending on its inclination to the direction of the incident light) light to the eye or not, appears to be brightly lit, at other inclinations appears to be dark.
You can observe this readily with a very thin, wedge-shaped plate, for example, the one shown in Fig. 750. The locations of greatest darkness and brightness lie parallel to the edge AA' of the wedge, because the cuts cb parallel to it are locations of equal thickness. - The effect of the layer thickness becomes very clear (Newton) when you place a very weakly curved convex lens AB with a focal lens of several metres on a glass plate GH (Fig. 751) and look at it from above: You will see concentric, circular, dark and bright rings about the point of contact (Fig. 752). The lens and the plate are separated from each other by a layer of air, the thickness of which at the point of contact E is zero and increases very slowly outwards like for the wedge-shaped plate. The air takes the place of the thin plate. The locations of equal thickness are at the same distance from E, that is, they form circles around E; the thinner or thicker is the layer of air, the smaller or larger is the associated circle (Newton's rings). When the angle of incidence changes, also the radii of the rings change.
You can measure the wave length of light by means of Newton's rings: It can be computed from the radii of the lens and of the rings. In practical optics, one checks during the grinding of lenses with their help (Fraunhofer, Löber 1830-1912) to discover whether the surface to be ground has the same curvature as the test surface. If you place both on top of each other - one is convex, the other concave - you see rings, where the curvatures differ, and they disappear where they agree at all points.
The interference phenomena of mono-chromatic light on thin plates, plane parallel as well as wedge-shaped ones, discussed above only dealt with changes in brightness. Overall, the destruction or weakening of light at a location indicates destruction or weakening of the illumination. It is different in daylight. Daylight (white light) contains waves of all lengths, to which your eye reacts. If the thin plate (Fig. 749) is illuminated by white light and the angle of incidence and thickness of the plate are such that it looks black in homogeneous red light, the red will also disappear in the white light; however, only the red light disappears, while the other colours contained in the white light remain and the entire plate assumes the colour, which is the result of the remaining colours, that is, in a colour, which depends on the thickness of the plate and the angle of incidence of the light. If you turn it back and forth, it appears in turn in different colours; if it does not have parallel faces, it displays without you turning it at the differently thick locations different colours.
The same
explanation applies when the bands in wedge-shaped plates and
also in Newton's rings are not only intermittently
dark and bright, but have different colours, when they arise from
white light. Only the edge of the wedge and the point of contact
between the lens and the plate are black. All waves are then
destroyed by interference, that is, the day light is
extinguished. - Why are not
also thick plates coloured in daylight? Newton's rings answer the question: If the rings arise from
mono- chromatic light, they are still noticed far from the point
of contact of the lens with the plate, that is, where the air
layer is already quite thick. However, the further away they are
from the vertex of the lens, that is, the thicker the layer,
during the illumination of which arises interference, the closer lie the rings together and eventually
merge into each other. Hence, if you employ white light (with all
colours) instead of mono-chromatic light, there arise at that
location rings of all possible colours close together, so that
their mixture gives the impression of white - a lack of colour. A
similar explanation applies to the lack of colour of a thick
transparent plate in daylight. However, if you apply mono-chromatic light, you can still detect also in the
case of considerable thickness of a plate interference of the
light, indeed it becomes a criterion for whether a given plate of
glass, which is 
supposed to
have parallel faces, really has parallel faces. If it does not, you see lines of interference (Fizeau bands) for a similar reason for which a wedge-shaped
plate exhibits them. However, it is much harder to detect the
interference lines and at a certain thickness of the plate they
also disappear in mono-chromatic light. Indeed, they appear also
in perfect plates with parallel faces (Haidinger 1795-1871 1854), but they demand special experimental
conditions to become visible (Lummer). If these
conditions are met, they yield Haidinger's rings,
the base for interference
spectroscopy, by far
the most sensitive method for the examination of a source of
light - in particular, a spectral line - for its homogeneity (Charles Fabry-Perot 1867-1945 Air plate spectrocope; Ernst Gehrke 1878-1960 Glass plate spectroscope).
Application of interference
phenomena to physical measurements
Interference
curves are used in the inference
refractometer of Jules C. Jamin 1818-1886, in order to detect very
small changes in refraction indices, and in the Fizeau-Abbe dilatometer in order to measure thermal expansion
coefficients of solids. The former employs the bands of Brewster. If you
place two equally thick, transparent plates with
parallel faces facing each other and let parallel rays of light
meet them, then during its passage the light is refracted and
reflected several times (Fig. 753). Since the plates have exactly the same thickness and
have exactly parallel faces, parallel rays leave the plates at
equal phase and also arrive like this at the eye, because what
applies to a single ray is valid for all, since there cannot
arise a difference. However, if the plates are not exactly
parallel to each other, the ray paths differ from each other,
whence they can interfere with each other. If you then look
through the plates at the source of light, you detect certain
interference curves (Brewster's bands). The set-up of Fig. 754 does not show them very clearly, Jamin's refractometer much better (Fig. 755). You see the incident ray is split by
refraction and reflection (the plates P1 and
P2 are silver-plated on their back
sides) into two rays 2 and 3. The two rays
which eventually interfere are very far apart. 
This is one of the main advantages of
the instrument; which has been enlarged further by special tricks
(Mach and Ludwig Albert Zehnder) to 50 cm. Fig. 750 displays how this is employed in the instrument. For
example, Jules C. Jamin has investigated with it the ratio of
refraction of air at different temperatures. You bring two
exactly identical tubes, closed by glass plates, into the path of
the rays of light. As long as they agree, the difference in the
paths of the rays through the tubes remains unchanged, that is,
also the interference image. However, the slightest change in one
of the tubes causes displacements of the interference bands. You
observe them from A with a. telescope.
You employ the Fizeau-Abbe dilatometer for measurements of thermal expansion coefficients of solids. The principle is due to Fizeau, the instrument and several improvements of the initial method of observation due to Abbe. You use the interference curves, shown by a V-shaped, very thin layer in parallel mono-chromatic light.The essential part of the instrument is Fizeau's table (Fig. 756). The steel table top T carries the object O to be measured with almost parallel lens faces. Close above lies the glass plate P with parallel faces, the distance of which from the object is controlled by screws. You make the space between them wedge-shaped and smaller than 1/10 mm and illuminate the wedge of air from above by parallel mono-chromatic light. There arise interference bands, parallel to the edge of the wedge. If the object expands, (the expansion of the screws is specially allowed for), the thickness of the wedge changes and the curves move. You observe the bands through a telescope and employ their micro-metrically measured displacement with respect to the mark m in the computation.
Very important instruments are the interferometers of Michelson. Initially only the arrangement in Fig. 748 was called an interferometer, however, depending on the task it had to solve and the corresponding induction of interference, he renamed it. Fig. 748 shows the Michelson experiment (1883), which was to determine whether Earth travels through light's ether (ether wind, ether drift) or whether it takes it with it, in other words: Whether the light's ether takes part in Earth's motion or not. It ended up with the question whether light spreads faster in the direction of Earth's motion than perpendicularly to it, that is, the question, whether the light from A-I and back moves faster when the segment A-I is in the direction of Earth's motion than when it is perpendicular to it. If the experiment confirmed the question, the interference bands during the first position of the interferometer would have to lie differently than during the second; they had to displace perpendicularly to their length as the interferometer was rotated in its plane by 90º. The experiment yielded negative results.
In order to still increase the accuracy of his measurements, Michelson proceeded to the set-up shown in Figs. 757 and Fig. 758. The result was the same*. We have discussed earlier the significance of the negative result of Michelson's experiment: It is the indispensable assumption for the theory of relativity. Best known among the applications of interferometers to measurements has become the calibration of the metre in lengths of waves of light, moreover a new form of Fizeau's experiment and finally measurements of angles in astronomy, which hitherto could not be performed due to the smallness of the quantities to be measured, especially the measurement of the diameters of several stars and the mutual distances of double starts in squares.
*The experiment was repeated with the same result and yet larger accuracy by Joos (1930 in Jena, Germany, with an interferometer, built by Zeiss.
Diffraction of light through a
narrow slit (
Grimaldi 1665)
The
co-operation of the elementary waves explains, according to Fresnel 1819, why light, as a rule, propagates recitlinearly,
that is, in rays which do not bend around corners. Parallel rays
of light enter a dark room through a sharply bounded, vertical
slit (1/2 mm) in its shutter and meet at a large distance from
there (2 - 3 m) a white wall parallel to the window. We then
expect on it a sharply bounded, vertical, bright band of the
width of the slit. However, the wall (Fig. 759 a) displays a much wider band and on both sides of it several parallel narrow bands,
intermittently bright and dark, the bright ones already at a
short distance from the central band losing so much brightness
that they are soon undetectable and only darkness is seen. If the
light is mono-chromatic, the bands are mono-chromatic bright and
dark, otherwise they have mixed colours. Fig. 760 is a horizontal section through the
slit and the screen. S is the cut through the band of
light, which we alone expected, s1, s2,
s3 those through other bands. During
the passage through S, the rays have continued in their
direction of incidence, but are also refracted around the edges
of the slit. An analogue to this is: If you place in the path of
parallel rays of light a very small opaque body (a straight wire
in tension) and let it throw a shadow on a distant screen, the
shadow is not sharp, but is on both sides bounded by very fine,
intermittent light and dark bands. - This deflection of light is
called diffraction, the slit the diffraction opening. The form of the diffraction figure on the screen (a, b, c)
depends on that of the opening; in what follows, we will assume
the it is always a straight, very narrow slit, sharply bounded by
two edges - very narrow
means here: Not very
wide compared with the length of the waves of light.
How do these diffraction bends arise? Let the rays passing through the slit be parallel, that is, the source of light be infinitely far away from the diffraction opening. All ether particles in the plane of the slit are then seized simultaneously by the wave motion. Every point of the slit becomes the origin of a wave system (Huygens' principle). All wave systems form simultaneously, that is, all of them have the same phase, that is, the are capable of interference. Whether a certain point of the screen is lighted up or not depends on its distance from the slit, because it depends on it whether the waves, which meet at it, support or more or less annul each other. However, this depends on the angle, at which a bundle, emitted from the slit, leaves it. The figures shows everything very exaggerated; the location s, given as lines, indicate the points, at which intersect the almost parallel rays coming to them. The central point of each line s can be considered to be the joining point of the corresponding parallel ray bundle. If it is very far from the slit, compared with the width of the slit (about 1/2 mm), - say, about 2 m - then the lines from the split's edges to it may be considered to be parallel.
All the rays of the bundle s (Fig. 760), which continue in the direction of incidence of the light, cover from the slit S to their point of intersection equally long distances, that is, arrive at the same phase at the screen and therefore generate the bright bands s, the centre of the diffraction figure. It is different at the point s1, which lies by half a wave length closer to the one edge of the slit than to the other edge. Now, if there were emitted from each edge a wave, so that there existed only two coherent wave centres, then s1 would be dark, since one wave crest of the one system would be met by a wave trough of the other system. However, also the points in between the edges emit waves. Hence s1 is also met by waves, the phase difference of which is less than half a wave length, which therefore do not completely annul each other: At s1 exists a certain brightness. What applies to s1, is valid for every point on the vertical line through s1, since Fig. 760 is a horizontal section through the slit and the screen. This line is the first band, lighted up on its side. Its brightness is 0.4 that of the central, bright band.
The point s2, which lies by one entire wave length closer to the edge S2 than to the edge S1, is quite dark. The wave, which arrives at it from S1, differs from the corresponding one, coming from S2, by an entire wave length. As it arrives at s2, it differs from that coming from the centre of the slit, by half a wave length; hence those two waves annul each other as they meet on the screen. Correspondingly, there exists for each point between S1 and m one point by half a slit width from it between m and S2, so that this entire bundle annuls itself. Hence you have the result: Through those points, for which the difference of their distances from the slit's edges are an entire wave length, passes the first dark band. - We go now to the point s3, which lies to the one edge of the slit by three half wave lengths than to the other and imagine that the bundle 3 (Fig. 761) has been subdivided into three equal parts. The edge rays of pairs of neighbouring thirds of the bundle,
that is, the ray
from S1and that from m1,
moreover, the ray from m1 and that from m2,
moreover, the ray from m2 and that from S2,
differ each by half a wave length, whence two of them annul each other as they arrive at the location where they meet, and only the third, that is, only one third of the entire bundle survives. The meeting point s2, that is, the vertical line through it, has still some brightness, but only 0,045 of that of the central bright band. If the passage difference of the edge rays is 4l/2, this entire bundle annuls itself (just as that going to s2) again, the bundle with the passage difference 5l/2 yields again a bright band, etc. However, already the third bright band has only 0.016, that is, 1½ % of the brightness of the central image. The bright bands vanish at a short distance from the central one and only the shadow is visible. Hence the effect of the light is, as a rule, perceivable only in the straight continuation of the rays passing through the slit, the light which goes around the corner is not, as a rule, observed. - The formation of the rectilinear rays gave the opponents of the wave theory an effective argument. Its resolution has become one of the strongest proofs of the wave nature of light. The diffraction phenomena yield even the best method for the measurement of the wave length of light.
You distinguish between Fresnel's and Fraunhofer's diffraction phenomena. For example, a Fresnel one is the striation at the edge of a shadow, which a brightly lit needle throws on a screen as well as that one, by which we first have explained diffraction; for example, a Fraunhofer one is the striation, which you see when you view through a trimmed feather a distant source of light as well as the halo as you view a distant source of light (sun, moon, lamp) through foggy air - in this case, the little drops of water diffract - as well as the rainbow colours, which you see when looking at a spider web lit by the sun and - note! - the image under a microscope of an illuminated specimen. Hence you observe Fraunhofer diffraction phenomena by viewing a light source (with an armoured or mere eye), Fresnel's by intercepting light influenced by a diffracting slit on a screen. In order to generate Fraunhofer diffraction phenomena, you must have the light source as well as the eye infinitely far away from the diffraction opening, for example, by letting parallel rays of a source of light (Fig. 765) exit through a collimator tube L and (behind the diffraction screen) the diffracted, parallel rays of the bundle enter a telescope F, focussed at infinity. In order to generate Fresnel's diffraction phenomena, you take the source of light as well as the screen to any finite distance from the diffracting edges. Hence Fraunhofer's diffraction phenomena are limiting cases of Fresnel's phenomena.
The distance of each band from the centre of the diffraction figure depends on the wave length of the light (Fig. 762). The shorter (longer) is the wave, the shorter (longer) is the distance of the first bright band from the centre of the diffraction figure. - Or also: The angle w, at which the light bundle, belonging to to a given band, is diffracted away from the slit depends on the wave length. The longer (shorter) is the wave, the more pointed (less pointed) is the angle, which the bundle, going to the first bright band, forms with the centre of the slit. For small diffraction angles j (Fig. 763) - we are here concerned with angles of only a few minutes (the figure has been exaggerated for the sake of clarity) - the phase difference of the edge rays is proportional to the diffraction angle j . If the difference for j is 2/2·l, it is for 2j , 3j , ... 4/2l, 6/2 l, etc. Hence: If the distance of the 1. dark band from the centre is n, that of the 2., 3., ... from the centre is 2n, 3n, ..., whence the interval between two neighbouring dark bands is n. Between two neighbouring dark bands lie the bright ones, the ones on the side are equally wide, the central band is twice as wide as the side ones.
If the slot glows in red light, the first bright (red) band lies near red (Fig. 762), in violet light near violet, while in white light there arise bands in all colours red to violet, corresponding to the colours contained in white, simultaneously side by side: A diffraction spectrum forms (Fig. 764, III,1).
Measurement of wave length of light
Diffraction allows you to measure
wave lengths of mono-chromatic light: In Fig. 763, the parallel lines represent the bundle going to the
first dark band. In the triangle abc, ac/ab
= sin
abc,
where ac is the wave length l and ab
the width b of the slot, whence
l=b ·sinabc.
You can measure the width b and, since the abc is equal to j, you find
l by measuring j. For this purpose, you refine extraordinarily the
method of observation as far as it concerns the slot and screen.
You do not let the diffraction figure form first on the screen
before you observe it, but you let (following Fraunhofer) the parallel bundle coming from the slot enter a telescope,
directed at the slot. The diffraction figure then arises (because
the rays are parallel) in the focal plane of the objective and is
enlarged by the eye piece (Fig. 765). For this purpose,
you do not employ a single slot, but a grating.
Diffraction grating
If you employ two equally wide, parallel slots side by side, which are one width of a slot apart, you obtain the diffraction pattern of Fig. 766. Depending on the shape (circle, rectangle, triangle) of the diffracting opening and their number, another diffraction pattern is formed (in white light, it is coloured, since the maxima and equally so the minima of the different colours do not occur at the same locations). The diffraction grating is especially important for optical measurements. If you increase gradually the number of slots, the number of minima grows, they move closer together and the maxima become thereby narrower, the minima cover eventually completely the space between the maxima, which become brighter (this is linked to the number of grating openings). In essence, this is what happens as the number of slots is increased to form a grating: A grating is a multitude of microscopically narrow slots, separated from each other by equally small grating rods; straight lines engraved on a glass plate - 300 - 400 per 1 mm (first made by the optician Norbert in Greifswald in 1846) form the grating rods which bound the slots. If you view through a telescope, focussed at infinity, such a grating, illuminated by parallel mono-chromatic light, you see (Fig. 764 I and II) a row of sharp lines of light: Parallel to the grating rods, equally far apart and symmetrical with respect to the one at the centre and the less lighted up the further away they are from the centre. In white light, you obtain with a grating pure spectral colours (with a single slot only mixed colours). Fig. 764 shows that for red the bands are further apart than for violet, moreover that in white light there occurs a white band at the centre (because bands of all colours are superimposed there) and symmetrically to it a number of spectra. The first is alone and is therefore clean, most of the second (spectrum of second order) is clean, but there enters already into its red end the blue start of the third, and from then on all spectra mix. Hence thus expanded spectra (for example, III3 in Fig. 764) are useless for observations. Following Michelson, broad pure spectra are obtained by plane glass plates, arranged like a staircase (stepped grating).
You measure wave lengths with a
diffraction grating, when the relation l = b ·sina/m 
applies with m the order number of the spectrum. If you employ the first spectrum, when
the relation is l=b ·sina, where b denotes the grating
constant, that is, the distance of the
centres of adjacent slits. The most reliable measurements of wave
length, made by Rowland, yielded for the Fraunhofer lines the values:
| line | Å·E | line | Å·E | ||||
| A | 7594 | E | 5270 | ||||
| B | 6867 | F | 4861 | ||||
| C | 6563 | G | 4308 | ||||
| D1 | 5896 | H | 3968 | ||||
| D2 | 5890 | K | 3923 |
The location of a definite colour in the diffraction spectrum relative to the other colours and relative to the ends of the spectrum depends only on their wave length, in the refraction spectrum also on the nature of the refracting ( and colour dispersing) substance. Fig. 768 shows both spectra of sun light.
Apart from glass gratings, also metal gratings are employed, when the lines are engraved in a metal mirror (up to 1700 per 1 mm, Rowland). The engraved lines disperse the light, the spaces in between act as mirrors, whence there arises a diffraction spectrum as with transparent gratings (reflection grating). Special advantages arise in the use of concave gratings (Rowland 1882), weak spherical reflection gratings, engraved into the concave side of a spherical surface with a large radius of curvature (1 - 6.5 m). They collect themselves the diffracted ray bundles (in the focal plane), that is, allow to omit the telescope objective and expand therefore the spectra far beyond the ultra red and ultra violet.
In order to observe and measure
spectra, formed with plane gratings,
like in the prism
spectroscope, you use a collimator tube and telescope (or camera)
- for a reflection grating both on the same side of the grating,
for a transparent one on opposite sides (Fig. 765). The concave grating demands a special set-up. In
order to present simultaneously at the largest possible dispersion the complete spectral
range in all accessible diffraction orders, you use the grating
set-up of Runge and Paschen (Fig. 769). You place on the circular table a large number of
photographic plates and record on them spectra of different
orders.
Certain natural colours arise through diffraction of light on grating like surfaces, for example, the colours of the wings of butterflies and other insects, which are covered with fine grooves, the colours of mother-of-pearl, pearls, etc. Brewster has reproduced the luster and colour of pearls by taking an impression of their surfaces in pitch and letting the light reflect at the grooves imprinted on the pitch.
Formation of image in microscope by diffraction (Abbe 1873)
The image of a normally illuminated object, formed by the objective of a microscope, is according the Abbe a Fraunhofer diffraction phenomenon. This object - a microscopic preparation - is almost never glowing by itself, it is mostly illuminated from a source of light by transmitted light*. (The condenser between the source of light and the object yields a bundle of light of intended opening and inclination.) While it passes from its source through the object, the light is diffracted like by a grating; however, the diffraction process generates as a diffraction pattern a number of spectra, which are congruous and lie symmetric about the centre of the slit. The objective forms this figure in its posterior focal plane. (You can see it, when you remove the eye piece of the microscope and view the focal plane of the objective with an auxiliary microscope.) In order to understand the image, which we view in a microscope through the eye piece, you must imagine the action of the image in the focal plane according to Huygens' principle transferred to the image plane, conjugate to the grating plane (Fig. 770). We cannot present this process here in detail and confine ourselves to the result: The refracted image comprises theoretically an infinity of spectra. In order to obtain an image, which is geometrically completely similar (correspondingly enlarged), you would have to include in this method, to be executed according to Huygens' principle, an infinity of spectra. In reality, there forms only a finite number of spectra; moreover, the objective can only seize a limited number, the more it seizes, the more similar becomes the image of the object. However, since the spectra become outwards very quickly darker, only a small number need pass through the objective, in order to produce a practically sufficient similarity between the object and its image. If beside the bright centre only one spectrum passes on each side, there arise still the bright and dark places at the right locations, although not in full sharpness. However, if there passes only the main maximum - the undiffracted image - there remains nothing which is characteristic for the object,whence you see only uniform illumination of the field. It can be shown experimentally that the image of the microscope is really formed.
The formation of the image by diffraction also explains why an as wide as possible opening angle of the objective is the main condition for a microscope's performance. The smaller the microscopic object, the more of its parts drop to small multiples of the wave length or even below one wave length, the further remove diffracted rays themselves of yet appreciable strength from the direction of the direct ray. Abbe wrote in his book "The optical tools of microscopy": "Therefore with decreasing dimensions an ever increasing opening angle of an optical interment is required, in order to be able to record yet the entire diffraction bundle to the border of minimal intensity; and in the end, also at the largest possible opening angle of our microscopes, only a small part of the diffracted light - the central part of the complete diffraction phenomenon - is accessible. The smaller this part, the further away is the microscopic image from a mere projection of the actual structure, the more disappear in it the individual characteristics of the object ··· Hence the aperture angle of the microscope is that element of the construction. by which the mapping process is influenced in its basic conditions. Its size determines the absolute measures, down to which material images yet admit a still complete, that is, a conformal mapping of the object's conformal state". In addition, this explains the importance of the numerical aperture and the significance of the immersion systems for the performance of the microscope and eventually altogether the limit of its performance.
Also the physical mapping of opaque (diffusely reflecting) objects rests on diffraction: The border of the structural parts of an object to be mapped diffracts the rays. The larger the diffracting parts with respect to the length of the light waves, in the narrower angular space move the diffracted rays. Accordingly, as long as the dimensions of the structural components (of whatever form and arrangement of parts) amount yet to considerable multiples of the wave length, all diffracted light of appreciable intensity is compressed into a small angular space about the direction of the direct (undiffracted) ray. In such a case, a small opening angle of the optical system suffices in order to make the entire light act and meet the conditions of a conformal mapping. On this rests, for example, that the eye can generate even of objects, which are very small, complete conformal images, provided only that they are large compared with the light waves (and just so the magnifying glass and telescope).
Abbe has explained by means of the diffraction theory of the formation of the microscopic image why you cannot raise the performance of a microscope arbitrarily by merely increasing the enlargement - in essence, because during mapping a point of light, due to diffraction, does not yield again a point, but a small disk. Thus, if two light points are so close together, that the two diffraction disks overlap, you cannot any longer perceive these points as separate ones, but only blended one into the other - whence there arises the limit of the performance of the eye, the microscope and the telescope. According to Abbe and Helmholtz, objects, which are smaller than half a wave length, can no longer be resolved. This limit has been substantially expanded by H.Siedentopf and R.Zsigmondy 1903 by a method, which makes ultra-microscopic particles visible: As a consequence, you can confirm the presence - not the true form - of particles; the diameter is approximately 40Å-E. The visualization of the particles rests on the generation of their diffraction disks. The particles are illuminated enormously strongly with the aid of a special condensers (sidewards, that is, perpendicularly to the tubus, by sun- or arc-light). As a result of their small size, they diffract the light and surround themselves with diffraction disks - circles of light. We see these circles , the shape of which does not depend on the true shape of the ultra-microscopic particles (whence they do not yield geometrically similar images) bright on a dark background just as we see dust in sun light. It is only assumed that the particles are so far apart that the microscope renders the disks separately. A main demand is: Only the diffracted light must enter the microscope and not a trace of the illuminating light. - By using the ultra method, you can distinguish in the microscope, for example, between media, which are completely homogeneous and therefore optically empty, and media, which appear to be opaque by more or less fine suspensions (colloidal solutions, ruby glass coloured by gold particles); you can study micro-chemical precipitation phenomena, during which a heterogeneous structure arises from an initially homogeneous one; very clearly the Brownian molecular motion of fluids, etc.
