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Diamond Design - Optical

It is to light, the play of light, its reflection and its refraction, that a gem owes its brilliancy, its fire, its colour.  We have therefore to study these optical properties in order to be able to apply them to the problem we have now before us :  the calculation of the shape and proportions of a perfectly cut diamond.

Of the total amount of light that falls upon a material, part is returned or reflected ; the remainder penetrates into it, and crosses it or is absorbed by it.  The first part of the light produces what is termed the " lustre " of the material.  The second part is completely absorbed if the material is black.  If it is partly absorbed the material will appear coloured, and if transmitted unaltered it will appear colourless.

The diamonds used as gems are generally colourless or only faintly coloured ; it can be taken that all the light that passes into the stones passes out again.  The lustre of the diamond is peculiar to that gem, and is called adamantine for that reason.  It is not found in any other gem, although zircon and demantoid or olivine have a lustre approaching somewhat to the adamantine.

fig. 12:  'cabachon'

In gem stones of the same kind and of the same grade of polish, we may take it that the lustre only varies with the area of the gem stone exposed to the light, and that it is independent of the type of cut or of the proportions given to the gem (in so far as they do not affect the area) ; this is why gems where the amount of light that is reflected upon striking the surface is great, or where much of the light that penetrates into the stone is absorbed and does not pass out again, are frequently cut in such shapes as the cabochon (fig. 12), so as to get as large an area as possible, and in that way take full advantage of the lustre.

In a diamond, the amount of light reflected from the surface is much smaller than that penetrating into the stone ; moreover, a diamond is practically perfectly transparent, so that all the light that passes into the stone has to pass out again.  This is why lustre may be ignored in the working out of the correct shape for a diamond, and why any variation in the amount of light reflected from the exposed surface due to a change in that surface may be considered as negligible in the calculations.

The brilliancy or, as it is sometimes termed, the " fire " or the " life " of a gem thus depends entirely upon the play of light in the gem, upon the path of rays of light in the gem.  If a gem is so cut or designed that every ray of light passing into it follows the best path possible for producing pleasing effects upon the eye, then the gem is perfectly cut.  The whole art of the lapidary consists in proportioning his stone and disposing his facets so as to ensure this result.

If we want to design a gem or to calculate its best shape and proportions, it is clear that we must have sufficient knowledge to be able to work out the path of any ray of light passing through it.  This knowledge comprises the essential part of optics, and the laws which have to be made use of are the three fundamental ones of reflection, refraction, and dispersion.

Reflection

Reflection occurs at the surface which separates two different substances or media ; a portion or the whole of the light striking that surface is thrown back, and does not cross over from one medium into another.  This is the reflected light.  There are different kinds of reflected light according to the nature of the surface of reflection.  If that surface is highly polished, as in the case of mirrors, or polished metals or gems, the reflection is perfect and an image is formed.  The surface may also be dull or matt to a greater or smaller extent (as in the case of, say, cloth, paper, or pearls).  The reflected light is then more or less scattered and diffused.

It is the first kind of reflection that is of importance to us here, as diamond, owing to its extreme hardness, takes a very high grade of polish and keeps it practically for ever.

fig. 13: Reflection experiment

The laws of reflection can be studied very simply with a few pins and a mirror placed at right angles upon a flat sheet of paper.

A plan of the arrangement is shown in fig. 13. The experiment is as follows :--

  1. A straight line A B is drawn upon the paper, and the mirror is stood on the paper so that the plane of total reflection (i.e. the silvered surface) is vertically over that line.  Two pins P and Q are stuck anyhow on the paper, one as near the mirror and the other as far away as possible.  Then the eye is placed in line with P Q at 1, so that Q is hidden by P.  Without moving the eye, two more pins R and S are inserted, one near to and the other far from the mirror, in such positions that their images appear in the mirror to lie along P Q continued.

    If the eye is now sighted from position 2 along S R, Q and P will appear in the mirror to lie on S R continued.

    The mirror is now removed, P Q and S R are joined and will be found to intersect on A B at M. If a perpendicular M N be erected on A B at that point, the angles N M P and N M S will be found equal.

    The above experiment may be repeated along other directions, but keeping the pin S at the same point.  The line of sight will now lie on P' Q', and the angles between P' Q', S R' and the normal will again be found equal.

    In the first experiment S appeared to lie on the continuation of P Q, in the second it appears to be situated on P' Q' produced.  Its image is thus at the intersection of these  two lines, at L.  It can easily be proved by elementary geometry (from the equality of angles) that the image L of the pin is at the same distance from the mirror as the pin S itself, and is of the same size.

  2. If the pins P, Q, R, S in the first experiment be placed so that their heads are all at the same height above the plane sheet of paper, and the eye be placed in a line of sight with the heads P, Q, the images of the heads R, S in the mirror will be hidden by the head of pin P.

    The angle N M P (position I) is called the angle of incidence, and the angle N M S angle of reflection.

The laws of reflection (verified by the above tests) can now be formulated as follows :--

  1. The angle of reflection is equal to the angle of incidence.
  2. The paths of the incident and of the reflected ray lie in the same plane.

From I it follows, as shown, that

  1. The image formed in a plane reflecting surface is at the same distance from that surface as the object reflected, and is of the same size as the object.
Refraction

When light passes from one substance to another it suffers changes which are somewhat more complicated than in the case of reflection.  Thus if we place a coin at the bottom of a tumbler which we fill with water, the coin appears to be higher than when the tumbler was empty ;  also, if we plunge a pencil into the water, it will seem to be bent or broken at the surface, except in the particular case when the pencil is perfectly vertical.

fig. 14:  Refraction experiment

We can study the laws of refraction in a manner somewhat similar to that adopted for the reflection tests.  Upon a flat sheet of paper (fig. 14) we place a fairly thick rectangular glass plate with one of its edges (which should be polished perpendicularly to the plane of the paper) along a previously drawn line A B.  We place a pin, P, close to the edge A B of the glass plate and another, Q, close to the further edge.  Looking through the surface A B, we place our eye in such a position that the pin Q as viewed through the glass is covered by the pin P.  Near to the eye and in the same line of sight we stick a third pin R, which therefore covers pin P.  The glass plate is now removed.  P Q and P R are joined, a perpendicular to A B, M M', is erected at P, and a circle of any radius drawn with P as centre.  This circle cuts P Q at K and R P at L.  L M and K M' are drawn perpendicular to M M', L M and K M' are measured and the ratio (L M) / (K M') found.

The experiment is repeated for different positions of P and Q and the corresponding ratio (L M) / (K M') calculated. It will be found that for a given substance (as in this case glass) this ratio is constant.  It is generally called the index of refraction, and generally represented by the letter n.

Referring to fig. 14, we note that as

P K = P L = radius of the circle,

we can write

                  L M
                 -----
    L M           P L           sin R P M
   -----   =   ---------   =   -----------
    K M'          K M'          sin Q P M'
                 -----
                  P K

Writing the angle of incidence R P M as i, and the angle of refraction Q P M' as r, this equation becomes
        n  =  (sin i) / (sin r)    (Eq. 1)

or
  n sin r  =   sin i               (Eq. 2)

In this case the incident ray is in air, the index of refraction of which is very nearly unity.  With another substance it can be shown that equation (2) becomes
  n sin r  =  n' sin i             (Eq. 3)

where n' is the index of refraction of that substance.

It can be seen easily, and in a way similar to that used with reflection (i.e. sighting along the heads of pins), that in refraction also :

The paths of the incident and of the refracted ray lie in the same plane.

fig. 15: Refraction bends light.  Too much bending causes total reflection instead.

Of two substances with different index of refraction, that which has the greater index of refraction is called optically denser.  In the experiment the light passed from air to glass, which is of greater optical density.  Let us now consider the reverse case, i.e. when light passes from one medium to another less dense optically.  Suppose a beam of light A O (fig. 15) with a small angle of incidence passes from water into air.  At the surface of separation a small portion of it is reflected to A" (as we have seen under reflection).  The remainder is refracted in a direction O A' which is more divergent from the normal N O N' than A O.

Suppose now that the angle A O N gradually increases.  The proportion of reflected light also increases, and the angle of refraction N' O A' increases steadily and at a more rapid rate than N O A, until for a certain value of the angle of incidence B O N the refracted angle will graze the surface of separation.  It is clear that under these conditions the amount of light which is refracted and passes into the air is zero.  If the angle of incidence is still greater, as at C O N, there is no refracted ray, and the whole of the light is reflected into the optically denser medium, or, as it is termed, total reflection then occurs.  The angle B O N is called the critical angle, and can easily be calculated by (3) when the refractive indices n and n' are known.  It will be noted that when the angle of incidence attains its critical value i', the angle of refraction becomes a right angle, i.e. its sine becomes equal to unity.

Substituting in (3),

        n sin r  =  n' sin i'
          sin r  =  1

                     n
          sin i' =  ---                    (Eq. 4)
                     n'

Or, if the less dense medium be air,

              n  =   1

                     1
          sin i' =  ---                    (Eq. 5) [8]
                     n'
Note 8.  No mention is made here of double refraction, as the diamond is a singly refractive substance, and it was considered unnecessary to introduce irrelevant matter. 

This formula (5) is very important in the design of gems, for by its means the critical angle can be accurately calculated.  A precious stone, especially a colourless and transparent one like diamond, is cut to the best advantage and with the best possible effect when it sends to the spectator as strong and as dazzling a beam of light as possible.  Now a gem, not being in itself a source of light, cannot shine with other than reflected light.  The maximum amount of light will be given off by the gem if the whole of the light that strikes it is reflected by the back of the gem, i.e. by that part hidden by the setting, and sent out into the air by its front part.  The facets of the stone must therefore be so disposed that no light that enters it is let out through its back, but that it is wholly reflected.  This result is obtained by having the facets inclined in such a way that all the light that strikes them does so at an angle of incidence greater than the critical angle.  This point will be further dealt with in a later chapter.

The following are a few indices of refraction which may be useful or of interest :--

  Water              1.33
  Crown glass        1.5  approx.
  Quartz             1.54 - 1.55
  Flint glass        1.576
  Colourless strass  1.58
  Spinel             1.72
  Almandine          1.79
  Lead borate        1.83
  Demantoid          1.88
  Lead silicate      2.12
  Diamond            2.417 (Eq. 6) [9]

These indices have, of course, been found by methods more accurate than the tests described.  One of these methods, one particularly suitable for the accurate determination of the indices of refraction in gems, will be explained later.

With this value for the index of refraction of diamond, the critical angle works out at

             1
  sin i  =  ---
             n
               1
         =  -------  = 0.4136
             2.417
      i  =  arcsin(0.4136)
      i  =  24 26'                  (Eq. 7)

This angle will be found very important.

Dispersion

What we call white light is made up of a variety of different colours which produce white by their superposition.  It is to the decomposition of white light into its components that are due a variety of beautiful phenomena like the rainbow or the colours of the soap bubble -- and, it may be added, the " fire " of a diamond.

The index of refraction is found to be different for light of different colours, red being generally refracted least and violet most, the order for the index of the various colours being as follows :--

Red, orange, yellow, green, blue, indigo, violet.

Note 9.  In the list given above the index of refraction is that of the yellow light obtained by the incandescence of a sodium salt.  This colour is used as a standard, as it is very bright, very definite, and easily produced. 
fig. 16:  Dispersion -- glass plates have no fire.

If white light strikes a glass plate with parallel surfaces (fig. 20) the different colours are refracted as shown when passing into the glass.  Now for every colour the angle of refraction is given by (equation (2))
  n sin r  =  sin i

When passing out of the glass, the angle of refraction is given by
  n sin i'  =  sin r'

As the faces of the glass are parallel, i'  =  r.

Therefore, r'  =  i, and the ray when leaving the glass is parallel to its original direction.  The various colours will thus follow parallel paths as shown in fig. 16, and as they are very nearly together (the dispersion is very much exaggerated), they will strike the eye together and appear as white.  This is why in the pin experiments on refraction, dispersion was not apparent to any extent.

fig. 17:  Dispersion -- prisms have fire.

If, instead of using parallel surfaces as in a glass plate, we place them at an angle, as in a prism, light falling upon a face of the prism will be dispersed as shown in fig. 17 ;  and, when leaving by another face, the light, instead of combining to form white (as in a plate), is still further dispersed and forms a ribbon of lights of the different colours, from red to violet.  Such a ribbon is called a spectrum.  The colours of a spectrum cannot be further decomposed by the introduction of another prism.

The difference between the index of refraction of extreme violet light and that of extreme red is called dispersion.[10] Dispersion, on the whole, increases with refractive index, although with exception.  The dispersion of a number of gems and glasses is given below :--

Quartz             0.013
Sapphire           0.018
Crown glass        0.019
Spinel             0.020
Almandine          0.024
Flint glass        0.036
Diamond            0.044
Demantoid          0.057

Note 10.  Generally two definite points on the spectrum are chosen ;  the values given here for gems are those between the B and G lines of the solar spectrum. 
Editor's note 4 has more details. -- Ed.

The greater the dispersion of a medium, other things being equal, the greater the difference between the angles of refraction of the various colours, and the further separated do they become.  It is to its very high dispersion (the greatest of all colourless gem-stones) that the diamond owes its extraordinary " fire."  For when a ray of light passes through a well-cut diamond, it is refracted through a large angle, and consequently the colours of the spectrum, becoming widely separated, strike a spectators eye separately, so that at one moment he sees a ray of vivid blue, at another one of flaming scarlet or one of shining green, while perhaps at the next instant a beam of purest white may be reflected in his direction.  And all these colours change incessantly with the slightest motion of the diamond.

The effect of refraction in a diamond can be shown very interestingly as follows :-- A piece of white cardboard or fairly stiff paper with a hole about half an inch in diameter in its centre is placed in the direct rays of the sun or another source of light.  The stone is held behind the paper and facing it in the ray of light which passes through the hole.  A great number of spots of the most diverse colours appear then upon the paper, and with the slightest motion of the stone some vanish, others appear, and all change their position and their colour.  If the stone is held with the hand, its slight unsteadiness will give a startling appearance of life to the image upon the paper.  This life is one of the chief reasons of the diamond's attraction, and one of the main factors of its beauty.

Measurement of Refraction

In the study of refraction it was pointed out that the manner by which the index of refraction was calculated there, although the simplest, was both not sufficiently accurate and unsuitable for gem-stones.  One of the best methods, and perhaps the one giving the most correct results, is that known as method of minimum deviation.  Owing to the higher index of refraction of diamond it is especially suitable in its case, where others might not be convenient.

fig. 18: How to measure 'index of refraction' and 'dispersion'.

The theory of the method is as follows :--

Let A B C be the section of a prism of the substance the refraction index of which we want to calculate (fig. 18).  A source of light of the desired colour is placed at R, and sends a beam R I upon the face A B of the prism.  The beam R I is broken, crosses the prism in the direction I I', is again broken, and leaves it along I' R'.  Supposing now that we rotate prism A B C about its edge A.  The direction of I' R' changes at the same time ;  we note that as we gradually turn the prism, I' R' turns in a certain direction.  But if we go on turning the prism, I' R' will at a certain moment stop and then begin to turn in the reverse direction, although the rotation of the prism was not reversed.  We also note that at the moment when the ray is stationary the deviation has attained its smallest value.  It is not difficult to prove that this is the case when the ray of light passes through the prism symmetrically, i.e. when angles i and i' (fig. 18) are equal.

Let A M be a line bisecting the angle A.  Then I I' is perpendicular to A M.  Let R I be produced to Q and R' I' to O.  They meet on A M and the angle Q O R' is the deviation d (i.e. the angle between the original and the final direction of the light passing through the prism).

Therefore O I I'  =  d.

Draw the normal at I, N N'.
Then
  M I N'  =  I A M  =  a
if a be the angle B A C of the prism.

Now by equation (1)
  n  =  (sin i) / (sin r)
  i  =  N I R     =  O I N'  =  O I M + M I N'
     =  d + a =  (d+a)
  r  =  M I N'    =  a
therefore
  n  =  (sin (d+a)) / (sin a)  (Eq. 8)

fig. 19:  Spectroscope -- collimator, table, and telescope.

The index of refraction can thus be calculated if the angles d and a are known.  These are found by means of a spectroscope.  This instrument consists of three parts :  the collimator, the table, and the telescope.  The light enters by the collimator (a long brass tube fitted with a slit and a lens) passes through the prism which is placed on the table, and leaves by the telescope.  The collimator is usually mounted rigidly upon the stand of the instrument.  Its function is to determine the direction of entry of the light and to ensure its being parallel.  Both the table and the telescope are movable about the centre of the table, and are fitted with circular scales which are graduated in degrees and parts of a degree, and by means of which the angles are found.

Now two facets of a stone are selected, and the stone is place upon the table so that these facets are perpendicular to the table.  The angle a of the prism, i.e. the angle between these two facets, can be found by direct measurement with a goniometer or also by the spectroscope.  The angle d is found as follows :-- The position of the stone is arranged so that the light after passing through the collimator enters it from one selected facet and leaves it by the other.  The telescope is moved until the spectral image of the source of light is found.  The table and the stone are now rotated in the direction of minimum deviation, and at the same time the telescope is moved so that the image is kept in view.  We know that at the point of minimum deviation the direction of motion of the telescope changes.  When this exact point is reached the movements of the stone and of the telescope are stopped, and the reading of the angle of deviation d is taken on the graduated scale.

The values of a and d are now introduced in equation (8) :
  n  =  (sin (d+a)) / (sin a)  (Eq. 8)

and the value of n calculated with the help of sine tables or logarithms.

The values for diamond are
  n  =  2.417 for sodium light
  Dispersion  =  nred - nviolet = 0.044