**L6**** Light**

Refraction changes with the colour of the refracted light. In order to
simplify the following discussion, we will employ first of all
only mono-chromatic light, for example, yellow light which is
generated by cooking salt in the colourless flame of a Bunsen
burner (Fraunhofer's line *D*), and explain the
terminology and notation of refraction.

If you let the bundle of light *AB *(Fig. 656)
pass through water, made turbid by milk, - the drops of milk
display the track of the light like dust in air - the path of the
bundle *BC* in the water is not an extension of the bundle
*AB* in the air, but it forms an angle with it. (One part
of the light - *BR *- is reflected, but we will disregard
this fact.) You call this process *refraction*,
the surface *FF *(Fig. 657) separating the *media** *air and water the *refracting surface*, the ray *AB *the *incident ray*, *BC *the *refracted ray*, the perpendicular *BN* the *perpendicular of incidence*, *r *the *angle of refraction*, the plane in which lie the incidence
perpendicular and the incident ray the *plane of incidence* (here the plane of the drawing).

You relate the
direction of the ray to the perpendicular of incidence (like in
the case of reflection). The relationship between the incident
and refracted rays is the subject of the law of Snellius (about
1630). Also the *refracted* ray lies in the plane of incidence,
and, indeed, on the opposite side of the perpendicular of
incidence; the sin of the angle of incidence divided by the sin
of the corresponding angle of refraction is a fixed number *n*,
the magnitude of which depends on the nature of the adjoining
substances (here air and water) and the colour of the light:

*n*_{ab}
= sin* i*/ sin *r*.

The constant *n* does *not depend on the magnitudes of the
two angles **i **and** r*,
and *a* and* b *denote the two substances through
which the light passes. During transition of yellow light (Fraunhofer-line *D*)
from air (*a*) into water (*b*): *n*_{ab} = 4/3,
that is,

sin (angle of incidence in air)/sin (angle of refraction angle in
water) = 1.33, instead of which you write *n*_{D}*
= *1.33.

In
order to understand this equation, imagine that the intersection
of the incidence plane through the vessel filled with water is
circular (Fig. 568), the vessel is half full and the experiment
is such that a ray incident from the left hand side always meets
the centre; if you now erect the perpendicular of incidence *NN*,
then, for example, if *r* is is the radius of the circle, *e*_{1}*E*_{1}/*r* : *b*_{1}*B*_{1}/*r** = *4/3
and similar for *e*_{2}*E*_{2} and *b*_{2}*B*_{2},
etc. Hence the incident point *E*_{1 }or *E*_{2}
lies from the perpendicular *NN* 4/3 times as far as the
corresponding location *B*_{1 }or *B*_{2}, at which the refracted ray meets the
vessel in the water.

The number 4/3 is
called the *refractive
index* (-exponent,
-quotient, -coefficient, -number) of air with respect to water.
The following table presents the refraction index *n*_{D
}of air with respect to the stated substances:

glass (flint) | 1.54-1.8 | carbon disulphide | 1.6204 | |||

diamond | 2.4173 | oxygen | 1.000271 | |||

rock salt | 1.5443 | hydrogen | 1.000139 | |||

water | 1.3332 | nitrogen | 1.000298 | |||

alcohol | 1.3617 | air (free from CO_{2}) |
1.000293 |

Every colour of light has a different *n*,
whence you must state for every *n *the* *colour
involved (Maxwell's relation). Thus, *n*_{D} denotes the
refractive index for the yellow, corresponding to the Fraunhofer-line *D*.

The basic law of refraction comes from experiments, but it can also be derived from the wave theory of light. The law is confirmed by measurement of the refraction ratios at different angles of incidence, and especially by the agreement of the computed values and optical equipment, designed by means of it.

Moreover, experiments show: If
the light passes during its refraction the two substances in reverse order,
that is, it passes (Fig. 659) *first *through
water and *then** *through air, and the ray
travels in water (now as incident ray) along the same path from *B
*to *C*, which previously it had to travel as
refracted ray from *C* to *B*, it travels as
refracted ray in air from *C* to *E* which it
previously travelled as incident ray from *E* to *C*.
If the *refracted** *ray* CB* in Fig. 660
encounters a mirror *at
a right angle* at *B*,
it returns *to itself *and along the entire way along which it
came. This fact is referred to as the *principle of the reversibility of
the path of a ray**.*

During an inversion, the angle* r *becomes
the angle of incidence and the angle *i *the angle of
refraction, whence sin *r*/sin *i *= *n*_{b,a},
where *n*_{b,a }denotes the refractive
ratio during the passage of light from water into air. However,
from sin* i*/sin *r =* *n*_{a,b }follows
sin* r*/sin *i* = 1/*n*_{a,b},
whence *n*_{b,a} = 1/*n*_{a,b},
that is, if *n*_{air,water}* = *4/3, then *n*_{water,air}*
= *3/4.

These number apply for the transition of light
from *air** *into *water*, from *air* into *glass**, *etc.; however, they also yield the refractive
index between glass and water. In fact, *experiments tell us*: If *n*_{a,c} is the refractive
index from the substance *a* to *c *and *n*_{b,c}
that from *b *to the *same
**c*, then *n*_{a,c}/*n*_{b,c}
= *n*_{a,b}, that is, it is equal to the
refractive index from *a* to *b*. If *a *refers
to water, *b *to glass*, c *to air, then

*n*_{water,air}/*n*_{glass,air}
= *n*_{water,glass} or also (since *n*_{a,c}
= 1/*n*_{c,a})

*n*_{air,glas}/*n*_{air,water} = *n*_{water,glass}

For glass with *n *= 1.5,
you have *n*_{air,glass} = 3/2, *n*_{air,water}
= 4/3, whence *n*_{water,glass}=3/2·3/4=9/8, a
result, confirmed by measurements.

The relationship *n*_{a,c}/*n*_{b,c}
= *n*_{a,b} or, what is the same thing, *n*_{c,b}/*n*_{c,a}
=*n*_{ab}, reduces the number of
measurements which would have to be made to determine the
refractive ratios of substances *in pairs*: You measure the
refraction of *all* substances with respect to one - say, air, that is, you
measure the refraction into *all
*substances *from* air
and then *compute* all the other indices.

The refraction index for the
transition *from one
substance* into another is called *the relative index*; the *absolute* values apply to the transition of light from *empty space** *into a substance. If we replace the substance *c
*by vacuum and denote the absolute refractive indices from *a*
to *b *by *n*_{a }and *n*_{b},
then *n*_{a}/*n*_{b}
= *n*_{a,b}. The equation sin *i*/sin
*r *= *n*_{a,b} then becomes *n*_{a}·sin*
i=n*_{b}·sin* r*.

Let the wall (out
of glass) which light encounters as it spreads be *perfectly*
transparent. This means: The light, as it arrives at the surface
of the wall, passes through the *interface**
*between the glass and air *without**
*being weakened. However, this never happens, since one *part *of the light is *always *reflected, even at the surface of so
transparent substances as water and glass, into the substance
from which it has arrived. The surface of water reflects, so does
every pane of glass like the mirrors of shop windows. However,
the reflected light is always only a small part of the incident
light, while it becomes the larger, the more inclined the rays
are to the surface they encounter. Such reflected images have
therefore always very weak light. Their formation is always
accompanied by refraction, which is the *principal *part of the incident light passing
through the interface.

Note: During this
process, the light comes from air and *enters** *water or glass. However, it is
quite
different when it comes *from
*water or glass and *enters** *air. When a ray of light comes
from air and *enters* water or glass, the angle of refraction is *always* smaller than the angle of incidence;
however large is the angle of incidence (Fig. 661, left image,
between 0º and 90º), there always is added an angle of
refraction. However, it is different when the light comes *from** *water or glass and enters air.
The right image in Fig.661 explains: The angle of incidence* r
*in water is then largest when the angle of refraction in air
is a right one, because this is the largest angle, which the ray
can form with the perpendicular *NN *in the semi-circle of
air. This angle *r* is the *limiting angle*. The rays in Fig. 662, which form a still larger angle with the normal, for
example, *g*, *cannot
at all exit into the air*, they are reflected into the water, that is, they are
reflected at the interface of water and air. This process is
called *total reflection*, because *all *rays
are reflected, as you conclude from the fact that the reflected
light has the same strength as the incident light. In brief: If
light passes from a *more
strongly* refracting
substance in to a *more
weakly* refracting one,
that is, into a substance, in which the ray is refracted away
from the perpendicular and exceeds the angle of incidence by a
certain value, the light is reflected altogether back into the
more strongly refracting substance. - How large is the limiting angle? If *in
air *the angle of
incidence *i *is like in Fig. 661 (left image) a right
one, then sin 90º/sin*r* = *n*, and since

sin 90º = 1, sin *r* = 1/*n*.

The angle *r*
is the limiting angle: Imagine the sequence in which the ray
passes through the substances in the reverse order (Fig. 661
right image), you see then that *r* is the angle of
incidence, to which is added an angle of refraction: The emitting
ray *touches *the surface. For water, *n *=
1.33, whence the limiting angle is that angle the sin of which is
3/4, that is, the angle 48º 36' 25; for one kind of glass (light
crown glass ) with *n = *1.50, the limiting angle is 41º
48' 37|.

You can observe total reflection at a water surface which you view - as in an aquarium - from down below, but also by looking at it in a glass of water from below. You cannot see through the water surface and get the impression that it is a perfect mirror.

The total reflection of light
explains how certain objects which *as a rule** *are
shiny and at the same time transparent, become under certain
conditions *opque** *and dull, for example, powdered glass and ice,
snow, foam (irrespectively whether it is made out of coloured or
colourless substances). Powdered glass is, strictly speaking, a
mixture of glass and air. It is dull for the same reason as, for
example, polished silver is, when it is powdered: It turns its
reflecting faces in *all
possible directions*, that is, throws
light in all directions, that is, it is *diffusive*.
It is *opaque* due to the total reflection of light in the mixture of
glass and air; light cannot pass through the mixture. Powdered
glass becomes transparent, if you cover it with oil, which has
approximately the same refractive ratio as glass.

**Totally reflecting and
refracting surfaces as substitute for metal mirrors**

The optical industry employs totally
reflecting, refracting planes as substitutes for metal mirrors,
because they reflect much better. For example, Fig. 663
demonstrates its application at the hypotenuse of the
right-angled prism *A *or Fig. 740 at short sides of each of the two right-angled prisms.
You employ reflective prisms, in order to change systematically
the direction of the rays of light. Fig. 663 displays this for
the ray *s*, Fig. 740 for the ray with an arrow.
Occasionally, you employ several prisms for this purpose (Fig.
663) or you manufacture objects out of glass which correspond to
such action (Fig. 664).The first is done in the *camera lucinda* of Wollaston 1809, by
means of which you can draw natural objects and which you can
combine with a microscope.Fig. 663 shows the prism pair of the camera lucinda of Zeiss. The rays coming from the cusp reach your eye only by
the round about path through the camera lucinda, and indeed *out of the same final direction*, from which come also the rays from the
microscope; hence the eye projects the cusp on to the image seen
in the microscope. You view through the camera the paper, lying
next to the microscope, and the pencil on the image in the
microscope, with which you can draw the outlines. The rays from
the microscope do not pass through the prism, but pass its rim on
their way to your eye, in order not to be refracted by the prism.

Fig. 664 shows a
totally reflecting glass object, which corresponds to a
combination of *three* prisms: The equilateral, right-angled
prism *ABC*, bounded by two other right-angled prisms each
of which has a refracting angle of 30º. The first serves total
reflection, the two others prisms the refraction of the incident
and exiting rays. If the incident ray meets *AF* at *such *an angle of incidence that it after
refraction passes through *AB* *perpendicularly* - then you must have sin*a* /sin 30º = *n*, that is *a* = *n*/2 - then it also passes -
reflected in *AC *at 90º - also perpendicularly *BC*
and exits at the angle *a *from
*FE *(since the two 30º prisms lie symmetrically to the
two halves of the ray). In each of the two 30º prisms, the ray
passes the prism in the minimum orientation (cf. below). This is
important for the application of a glass body in a spectral
instrument. Every differently coloured light has another
refractive index *n*, corresponding to the relation sin *a* = *n*/2, that is, another angle*a* . You change *a* by turning the glass body and thereby
directing *AF* differently with respect to the incident
ray. You employ the body, because it diverts rays of all colours
by the *same* angle. A special design of spectral
apparatus is discussed below. You can then link the collimator
tube and the observation telescope rigidly and transit from one
colour to another by merely rotating the prism.

Lummer and Brodhun 1889 have
employed the fact that a perfectly transparent, but totally
reflecting bounding surface can replace a wall which is totally *opaque*, in order to replace the photometer
screen of Bunsen by a more perfect one. Bunsen's screen
is imperfect, because

**1.** the *opaque** *(not greased) part of the
paper - it is to reflect diffusely the incident light, but not
let any pass through - is *not
sufficiently opaque**
*(both its sides are therefore lit by the incident light as
well as somewhat by the passing light) and

2*.* the* **transparent** *(greased) part of the paper - it should not to
reflect *any** *of the incident light, that
is, let all of it pass - is *not
transparent enough*; it
reflect *a little*, whence you see both sides in a mixture
of both illuminations. Lummer and Brodhun have employed instead
of the paper screen a combination of totally reflecting glass
plates.

In Fig. 665, *A*
and *B *are two right-angled glass prisms, halves of a
glass cube. At *qh*, they are joined by putty with the same refraction index as their own.
Otherwise they are separated by air. *ll *and *ll *are are two diffuse radiating planes. The light
from *ll *is reflected totally at the air bounded
part of the hypotenuse of *B *towards *Y*; in
contrast, at the location *qh*, covered with putty,
nothing is reflected and all incident light passes through. The
light from *ll *, which meets *qh, *passes
completely towards *Y*, but at the air covered parts of
the hypotenuse of *A *it is totally passed on to *X*.
Hence an eye at *Y* sees *qh **only* in the light, which comes from *ll*,
the surrounding part *only
*in the light from *ll*.

The combination of
prisms in Fig. 665 only explains the basic idea of the photometer*
**screen*. Lummer and Brodhun have built
the photometer head in the forms *A *and *B *in
Fig. 666, which shows also the layout of the entire photometer.
The line *mn* represents the photometer bench, *m *and
*n *are the sources of light to be compared. In between
them, perpendicularly to the bench, is an opaque as white as
possible plate *ik*; one side of it receives only light
from *m*, the other only from *n*. The mirrors *f
*and *e* throw the light, which is diffusely reflected
from *l *and *ll *to
*A* and *B. *The prism *A *has here a
spherical surface and a circular area *rs *groiund on to
it. It is pressed firmly with this circular area against *B*
(without putty!).

If you view
through the telescope *W *the area *arsb*, you see
the image which you know from the Bunsen-photometer
(Fig. 667): A circular area *a*, surrounded by the
circular ring *b*, which is the more differently lit, the
more the lighting of the two sides of the plate *ik*
differs. At perfect *equality** *of this illumination, you see *a
*lit equally brightly as *b*, that is, a uniformly lit
circular area. In order to illuminate the circular area and ring
equally brightly, you shift the photometer screen along the bench
(Fig. 623). It is 2.5 to 3.5 times as sensitive as Bunsen's
photometer. The mean error of an adjustment remains below 1/2 %
during measurement of a 50 candle lamp, that is, below 0.25
Candle.

**Importance of the refractive
index**

Determination of the refractive index is one of the most important tasks of Optics. Microscope, telescope and photographic objectives owe their perfection to that of the art of melting glass, for which a knowledge of the index is indispensable. The refractive index of fluids has for Chemistry the same significance as other material constants; it frequently allows one to draw conclusions regarding the chemical structure of a body (molecular refraction). The purity of many fluids in use can be tested by means of their refractive index; for example, the purity of fats and volatile oils; milk is tested thus for its fat content, natural butter is distinguished from artificial butter.

There exist many methods and tools for the determination of the refractive index. We will describe two due to Ernest Karl Abbe 1840-1905: They form the basis for all later ones, also for those of Pulfrich 1858-1927, which are probably most commonly used. The spectrometer of Abbe serves the measurement with solid, transparent substances: It employs the principle of rays returning into themselves of Littrov 1781-1840. For the measurement with fluids, Abbe's refractometer is used: It employs total reflection.

**Method of rays which return into
themselves**

Fig. 668 explains
the determination of the refractive index with Abbe's spectrometer. *ABA*'*B*'*EF* is
the prism of Fig. 633 (the substance which has been given for the purpose of
the measurement this form), *bca* is a vertical section
through its *refracting** *edge *EF*, a *principal section* placed into the plane of the drawing; *FS
*is a ray of light and takes the place of a bundle of
parallel rays of light. We make the angle of incidence *i*,
at which *FS* meets *cb*, so large that the ray *after *refraction meets *ca **perpendicularly*. The corresponding angle of refraction
is *r*. Its two legs (ray and normal) are then
perpendicular to the two legs *ca* and *cb *of the
angle *d*. Hence *r *= *d*. Since then sin *i*/sin
*r* = *n* (the refractive index) and *r = d*,
it follows that sin *i*/sin *d* = *n*,
whence you measure the angles *i *and *d* for the
determination of *n*. We have already described the measurement of the refracting angle
of the prism. The *return
into itself* of the ray
coming out of the telescope can also be employed for the measurement of *i *(Figs. 659/660 above);
the ray *FST *returns into itself if it meets the plane *ac
*perpendicularly. Hence, in order to measure *i*, you
rotate the prism from the position, at which the signal of light
from *cb* returns into itself, until you note that the
light from the plane *ca *returns into itself and read off
the angle on the fixed graduated circle (Fig. 635).

**Method of minimum deflection of
a refracted ray**

A method proposed
by Fraunhofer is important: A bundle of parallel
(mono-chromatic) rays enters the space (Fig. 669, seen vertically
from above), horizontally and meets the prism with vertical,
refracting edge. The bundle passes through the prism and - you
should imagine that it is a pointer with end point *F *-
and draws on the opposite wall a bright spot, which, as a result
of refraction, does not lie on the extension of the incident
bundle at *X*, but is deflected towards the base (to *F*_{1}) by the
angle *s*. If you turn the prism about the vertical axis
through *a*, the spot *F *moves on the wall. If
thereby the angle of incidence increases, as in Fig. 669 during
the transition to Position 2 of the prism, the spot *F *moves*
**towards the
refracting edge* to *F*_{2}.
Hence the direction of the emitting ray approaches that of the
incident ray, that is, the deflection by the prism *becomes smaller*. However, starting from a certain
certain position of the prism, the spot returns, that is, the
deflection becomes again larger. At this position of *return** *of the prism, the deflection
is smallest (*minimum
position*). The ray
forms here (we omit the proof) inside the prism *equal** *angles with the sides of the
prism (*V* and *W* in Fig. 670) and the deflection
angle *d* is related to the refracting angle *L* of
the prism and the refractive index:

*n = *sin½(*d
+ L*)/sin½*l*.

The
method is realized by
means of a spectrometer (Meyerstein 1808-1882) with* two* telescopes
(Fig. 671) for the incident and exiting rays. When in Abbe's method the ray returns to itself from the prism (Fig.
672), its inclination to the refracting plane is exactly the same
as when it passes at minimum deflection through a prism with
twice as large a refracting angle; hence also in Abbe's method,
the measurement occurs in the minimum position of the prism (at
minimum deflection, Fig. 673). - Fraunhofer's
method is suitable for laboratories, Abbe's for
workshops, both are equally accurate.

**Method of total reflection. ****Abbe****'s refractometer**

The refractive
index of a fluid is normally determined by the method of total
reflection. Abbe's *refractometer** *involves for this purpose two right-angled glass prims *A*
and *B *with *known* index *n*; they form together a right-angled parallelepiped (Fig.
674). You place in between the prisms a drop of the fluid to be
tested. The index *n *of
the glass must be larger than that of the fluid, whence you cut
the prisms from glass with index *n *= 1.75, that is, larger than that of most fluids. If you
now place the pair of prisms with the fluid between them in the
track of a bundle of light the light does not* always** *pass, indeed *not* when
the angle of incidence, at which the boundary glass/fluid faces
the light, reaches or exceeds the limiting angle of total
reflection. Then the light is totally reflected as it transits
from the more strongly refracting glass into the more weakly
refracting fluid. The bounding angle *g*, at which this occurs, is indicated clearly by the
refractometer (Fig. 676). This angle and the known
index *n *of the prisms yields the index of the
fluid: *n = **n *sin
*g*, for* **n *is the index air-glass, *n* that
of air-fluid, whence the index fluid-glass is* n*/*n *and for the bounding angle *g *of total reflection during the
transition of light from the glass into the more weakly
refracting fluid it is sin *g *= *n*/*n*. The scale of the refractometer, along
which you read the bounding angle *n*, already yields the magnitude of *n *sin *g* (since *n *is
constant), that is, you obtain directly the refractive index of
the fluid.

Fig. 674 explains
the basic idea of the refractometer. *A* and *B *are
two right-angled glass prisms, *C *is the fluid to be
tested, *F* is the lighted point from which rays reach the
prisms, indeed *parallel* rays (with the aid of the fixed lens *J*).
The rays pass through the (rotatable) pair of prisms *AB*
as long as the angle of incidence, at which they enter from the
(more strongly refracting) prism *A *the (more weakly)
refracting fluid *C, *is *smaller*
than the bounding value - however, they do *not* pass as soon as *AB* is rotated so far that this
angle is reached. If the rays at all exit, they are parallel and
have passed through the plane parallel body *AB, **they are** *parallel to each other. An eye
at *O *receives the light from *F* and sees it like
a far distant object. (Light reaches the eye only along the path
with arrows.) However, apparently you only see the source of
light as long as the light enters at an angle to the bounding
plane *C *which is smaller than the bounding angle of
total reflection; if you turn the pair of prisms in the direction
of a growing angle of incidence, the wall becomes opaque as soon
as the position corresponding to the bounding angle is reached -
it becomes dark. The instrument *measures*
the angle when this happens.

In the routinely
employed refractometer (Fig. 675), the lay-out is as follows: **1.
**The eye and the source of light have been interchanged.
The light passes *first* through the prisms and *then* through the lens. With reference to
Fig. 674, this means that the light comes from the right hand
side and *F *is the eye. This inversion of the sequence
does not change the result (according to the principle of
invertibility of the path of the ray). The eye sees in the tube
the source of light *O *extinguished when during rotation
of the prisms the angle of incidence reaches the limiting angle. **2.**
The rays of light meet the layer *C **in all possible *directions. But only those can pass, the
angle of incidence of which is *smaller**
*than the limiting angle of total reflection. If you have
turned the prisms to the position at which the ray, which falls
parallel to the axis of the telescope on *B*, meets *C*
at the *bounding** *angle, then one half of the
field of vision - we do not explain why - receives no rays
whatsoever and is dark, the other is bright, whence it looks like
shown in Fig. 676. The operation of the refractometer is now
obvious: You place a drop of the fluid to be tested between the
two prisms, apply with the mirror (homogeneous) light, look into
the tube, rotate the prisms with the knob until the field of
vision becomes like shown in Fig. 676 and obtain through the
magnifying glass the index. (If the light is *not** *homogeneous, the subdividing line has colours
and is faded. There is a special arrangement for removing the
colour).

You can display
directly, when white light passes through a prism, that the rays
corresponding to different colours are refracted differently and
that thereby form colour dispersion. However, colour dispersion is only *measurable *by the refraction indices for the
individual colours and the relations between them. The glass
industry has special interest in fining out the magnitude of
dispersion: The suitability of a glass for microscopes and
telescopes (especially avoidance of coloured edges around images)
is largely determined by its *path *of
dispersion.

Following Abbe, you specify the properties of kinds of glass by the
refraction indices for the lines *A*'.b *C, D, F, G',*
more recently also for the helium line *d* and the mercury
lines *e*, *g, h*. The path of dispersion is
characterized by the refraction differences for the sections *A'
- C, C - e, e - F, F - g, g - h *as well as the ratio of
these partial dispersions to the average dispersion between the
lines *C* and *F*.

You try to express
the relationship between a wave length *l *and the corresponding refraction ratio*
n*_{l} by means of a* **dispersion formula*. You confirm the correctness of the
formula by checking the computed *n*_{l} against the value obtained from a
spectrometer. The oldest formula, due to Cauchy 1836, for normally (in contrast to anomalously)
dispersing, isotropic transparent substances is *n*_{l }= *a + b*/*l*^{2}+ *c*/*l*^{4} + ···, where *a*,
*b*, *c *··· are constants which characterize
the substance and must *first
be determined**.*
For most substances, *n*_{l }= *a + b*/*l*^{2 }is sufficient and agrees
with measurements. Hence, if you measure with a given substance
for *two *definite wave lengths (colours), for example *l*_{C} and *l*_{F}, the refraction ratios and
compute from the equations for *n*_{C}* *and
*n*_{F}* *the constants *a* and *b*,
you can compute for *every*
given *l *the corresponding *n*_{l }with sufficient accuracy and till agree
sufficiently with a control measurement.

However, Cauchy's formula proves to be true only for wave lengths
between 0.4 and 0.8 *m *(within
the visible spectrum). If you compute from it the refraction
numbers for longer waves (in ultra-red), the formula is less
useful even near the visible spectrum and becomes less for longer
wave lengths, for example, for water, glass, quartz, fluor spar,
rock salt. Modern dispersion theory, which employs electro-magnetic theory of light yields results which are much closer to
experimental results. It explains refraction by interaction
between the ether and the molecules of a substance, respectively
the electrons linked to the molecules, that is out of the
influence of the molecules on the vibrations of the ether. The
simplest such dispersion formula is *n*² = *b*²*
*+ *M*_{1}/(*l*²*
- l*_{v}²)
+ *M*_{2}/(*l*²
- *l*_{r}²) (Eduard Ketteler 1836-1900 1893 and Helmholtz 1893), where *M*_{1 }and *M*_{2
}are depend on the nature of the dispersing substance and
interlinked closely to its dielectric constant (cf. below);* l*_{v} and *l*_{r }are are two wave
lengths, characteristic for the substance in the ranges ultra-violet and ultra-red. The substance *absorbs* them, because they excite the electrons
in the molecules (resonance); they can be determined exactly
experimentally. For many substances such as water, rock salt,
sylvine, fluor spar (Ernest
F. Nichols 1869-1924, Heinrich Rubens 1865-1922), the formula represents
dispersion exactly. For example, for quartz, it is insufficient
and demands a further extension.