Page 1 of 11Page 1 of 11
`
`
`
`GIA EXHIBIT 1002GIA EXHIBIT 1002
`
`Page 1 of 11
`
`GIA EXHIBIT 1002
`
`

`
`Copyright
`
`All rights reserved. Copyright in articles published in this fournal is the property of
`Taylor & Francis Ltd.
`
`Authors are themselves responsible for oblainingjiermission to reproduce
`copyright material from other sources.
`Authors are required to sign a form for agreement of the transfer of_copyrighta
`
`The appearance ola code before the copyright line at the bottom ofthe first page__ofan
`article in thisjournal indicates the copy right owner’s consent that copies ofthe article
`may be made for personal or internal use, or for the personal or internal use of those
`clients, in the USA, who are registered with the Copyright Clearance Center. This
`consent is given on the condition that the co pier pays the stated per-copy fee through
`the Copyright Clearance Center, lnc., 21 Congress Street, Salem, Massachusetts 0i9?0,
`for copying beyond that permitted by Section 10? or 108 of the US. Copyright Law.
`This consent does not extend to other kinds ofeopying, such as copying for general
`distribution, for advertising purposes, for creating new collective works, or for resale.
`Fees for past articles are the same as'those shown for current articles.
`
`1
`
`-u
`
`Page 2 of 11
`
`

`
`I Thlsma2rIalm:ybepndne¢'1edbvComruhtl:w(11t|e11U.S.Cod2)
`
`OPTICA ACTA, 1981, VOL. 28, N0. 6, 801-809
`
`A ray tracing study of gem quality
`
`A. HARDY, S. SHTRIKMAN and N. STERN
`
`Department of Electronics, The Weizmann Institute of Science,
`Rehovot, Israel
`
`(Received 6 February 1980; revision received 1 August 1980)
`
`Abstract. A computerized three-dimensional ray tracing technique is utilized
`to obtain the scattered pattern of light perpendicularly incident on the table of
`gem stones. It is then shown that a correlation exists between gem cuts considered
`best by the diamond trade and the refiection pattern within a cone angle of about
`60°.
`
`1.
`
`Introduction
`
`The optical properties. of gemstones in general and diamonds in particular are
`closely related to their ‘beauty’. The beauty of a gem is revealed through the light
`being reflectcd by the stone and depends primarily on the material’s index of
`
`refraction and the shape in which it is cut. It was not until the early 20th century,
`however, that an attempt was made to define mathematically the properties of beauty
`[1]. Till then the cutting was solely an art which was expressed by the broad
`spectrum of diamond designs, most of them no longer in use. Nowadays, the most
`popular is the so-called brilliant cut [1], which is a 58-facet, eightfold symmetric
`design. Using a simplified two-dimensional model, together with geometric and
`optical arguments, Tolkowsky [2] was able to justify the shape and number of facets
`of that cut. Furthermore, he defined a criterion for beauty which served to find the
`
`most desirable proportions of the stone’s facets. Tolkowsky’s proportions are still
`accepted by the Gemmological Association of America as the best ones proposed so
`far. Alternative, though similar, approaches by Johnson [3] and Eulitz [4] produced
`other sets of proportions. A technique developed by Rosch [5] and later used also by
`Elbe [6], to approach the problem experimentally, produced a spot diagram of the
`light scattered by a gemstone. Recently a laser technique has been utilized [7] to
`obtain reproducible pictures of the scattered light, in order to be able to identify
`uniquely individual cut diamonds, for purposes such as insurance, etc. This
`technique made visible many spots otherwise not perceivable and one could notice
`that some relation exists between the scattered spots and the quality of the diamond
`cut. Thus, we attempted a computer study of gem cuts [8], by simulating and ray
`tracing the scattered light. That program was motivated by a goal to define realistic
`and useful criteria for diamond beauty in order to take the ‘art’ out of the trade and
`
`set it on better, more easily controlled and measurable parameters. More recently,
`Dodson [9] took advantage of the computer ray tracing approach in order to evaluate
`three statistical parameters which he associated with the previously more intuitive
`definitions of beauty, namely brilliance, sparkliness and fire. As a consequence of his
`study, Dodson proposed some new cut designs which he considers to be comparable
`to the traditional one, [or even more attractive.
`It is the purpose of the present paper to outline the computer ray tracing metltpd
`of [8] and bring together some characteristic results in an attempt to correlate some
`
`01130 .W0‘l/8|/‘28Ub U8lll S02-0|! I
`
`|08|
`
`'l':Iy|nr & l"r.inL'i.~ Ltd
`
`Page 3 of 11
`
`

`
`802
`
`A. Hardy et al.
`
`commonly used cuts with the subjective cut quality criteria as determined by experts
`in the diamond trade. A general description of the computer model is given in§ 2. Its
`
`application to some selected cuts is discussed in § 3. Suggestions for further
`
`investigation in the light of our conclusions are presented in §4.
`
`2. The computer model
`The experimental set-up [7] simulated by our computer program is schemati-
`cally described in figure 1. The lens L has a twofold purpose: first, to transpose the
`
`source light S into a collimated beam of parallel rays which impinge on the diamond
`D, and secondly, to collect the reflected light on a screen SC in order to obtain the
`far-field pattern within practical distances. In the computer program, however, the
`lens is omitted and the far-field pattern is represented by the direction angles of the
`
`scattered rays. The first part of the program generates a diamond and determines its
`orientation with respect to the set of impinging parallel rays. Owing to the high
`
`degree of symmetry of most gems, their shape can be defined by a small number of
`independent parameters. For instance, in order to describe fully the round brilliant
`
`cut, such parameters as the crown angle, pavilion angle, the number of facets, etc, ate
`
`sufficient. The coordinates of all corners and their corresp ondence to facets are then
`determined. A computer drawing of such a diamond, based on the above mentioned
`parameters, is presented in figure 2. The second part of the program simulates the
`
`interaction of light with the gem. In the present work only a set of parallel rays,
`incident perpendicularly to the table of the brilliant, is considered. Furthermore, we
`
`assumed that the illuminating light is natural, i.e. unpolarized [10]. This enabled us
`
`to exploit the high symmetry of the problem and to reduce the amount of necessary
`
`calculations. The program, however, is capable of dealing with a case in which any
`direction of orientation is chosen and includes the treatment of diffused light which
`
`is represented by many sets of rays having an arbitrary direction of propagation.
`
`Each ray, from the incident set, is propagated until it intersects with one ofthe facets.
`There it is either reflected or refracted in accordance with Snell’s law and Fresnel's
`
`formulaef. The reflected part is stored while the refracted ray is continued inside the
`gem. The simulation of reflections and refractions goes on until the computed
`
`Figure 1. The experimental set-up simulated by the computer program. The light source is
`indicated by 3. SC is a screen, L a converging lens and D the diamond. In the present
`paper we only consider rays incident perpendicularly to the diamond’s table.
`
`|_
`
`C
`
`1‘ Rigorously. the reflected light is partially polarized even though the incident wave is
`unpolarized [10]. The effect becomes pronounced only after several reflections but then the
`ray’s intensity is rather low. To simplify the problem and minimize computer time this effect
`was ignored and the reflectivity at each facet was always assumed to be that of natural light
`[10].
`
`Page 4 of 11
`
`

`
`_/I ray tracing study of gem qua{i'r_r
`
`STRR FACETS
`
`BEZEL FACETS
`
`UPPER G IRDLE FACETS
`
`GiRDLE
`
`LOWER GIRDLE FACETS
`
`PAV|L1ON
`
`-..
`
`psmcupat Pawuou FACETS
`
`\cu LET
`
`Figure 2. The modern brilliant cut.
`
`POLYHED RON
`SHAPE GENERATOR
`
`PARALLEL RAYS
`GENERATOR
`
`RAY TRACING
`PROGRAM
`
`DIS PLAY Bu‘-INA LYSIS
`PROGRfi.M
`
`Figure 3. The computer flow-diagram.
`
`intensity of the ray drops below a predetermined value (typically 0-01 of the input
`intensity). The same process is then repeated with all other rays. Eventually, all
`scattered rays with the same direction of propagation are grouped together, taking
`into account their relative intensity, thus producing a far—fielcl pattern which is
`represented as a spot diagram on a spherical surface. It is worth noting that
`alternative patterns, indicating the origin of each scattered ray on a facet, can also be
`obtained. Thus, one can properly take account of the relative visibility of the various
`facets of the gem. A flow diagram of our program is presented in figure 3.
`
`3. Application of the ray tracing model
`Based on the computer ray tracing approach described in the previous section, we
`corfccntrate here on a study of various gem cuts. The computed reflection patterns
`are visually compared to each other and to patterns most widely in use at present. An
`empirical criterion to characterize the best cut is then conjectured.
`In figure 4 we show the effect ofpavilion angle variations on the reflection pattern
`of a brilliant cut. On the right hand side of the figure a projection of the brilliant cut
`under consideration is shown. On the left, the corresponding reflection pattern is
`given (within a cone angle of 60°), while the middle column indicates the power flow
`
`Page 5 of 11
`
`

`
`A. Hard): et al.
`
`\
`ooosa+ 2°01:
`1- ODDS 3:+'O 00.)?
`O U 3 5 9
`
`Shollow Cut
`
`\
`
`0.00| 1: 00972 OWE
`+'
`+ \
`no-{as door: cease
`
`D04 39
`‘* +0.oIun
`
`Figure 4. Brilliant cut diamonds (right column) with different pavilion angles and their
`computed reflection patterns (left column) within a scanning cone of 60°, as viewed
`from the diamond. The maximum view angles are indicated by circles. Due to the
`eightfold symmetry, the power scattered in the direction corresponding to the spots is
`only given for one-eighth of the scanning circle (central column), in units relative to the
`incident power. As expected, the central spot is the most intense.
`
`in these directions in units relative to the incident power. The reflection pattern in
`the second row corresponds to a brilliant cut with proportions considered as perfect
`(Tolliowsky’s cut). One immediately noticed characteristic is the relatively homo-
`geneous distribution of points within scanning cone angle (60‘‘) of that pattern. On
`the other hand, the distribution patterns of a brilliant cut with either shallow or deep
`pavilion are much less hornogenous as indicated in figure 4. The shallow cut has a
`more intense central spot with a wide dark region surrounding it. The effect is due to
`internal reflections since the only parameter changed is the pavilion angle. As we
`change a well-proportioned cut into a shallow one, rays which otherwise would
`emerge obliquely out of the table now undergo multiple reflections from one
`principal pavilion facet to the table, back to the opposite principal pavilion facet and
`then perpendicular to the table, thus producing the for-field pattern, as mentioned
`above, for a wide range of shallow cuts. It is worth noting, however, that the near-
`field (on the diarnond’s surface) pattern of the shallow cut of figure 4, also has wide
`dark regions near the centre. Rays which perpendicularly enter the table generally
`leave near the edges, with only a negligible portion of their energy left for the near-
`eentral part of the gem. .This near-field pattern was previously observed and is
`known as the fish-eye effect in shallow stones [1]. The far-field pattern on the bottom
`of figure 4 corresponds to a brilliant cut with a deep pavilion. It is characterized by a
`very small number of points which are located near the centre. This should not
`
`Page 6 of 11
`
`

`
`A ray tracing study of gem quality
`
`805
`
`surprise us since the deep cut of figure 4- is close to a corner-cube shape, thus all rays
`
`are scattered back along the same path they were travelling when incident to the
`
`stone. With perpendicularly impinging rays, the ones emerging also propagate
`
`perpendicular to the table as indicated in figure 4. The only scattered rays that can
`enter an observe:-’s eyes are those illuminating the stone from the observer’s
`
`direction, usually from behind him. Such rays, however, are blocked by the
`
`observer's head, giving him the impression ofa gem with a dark table. That is why in
`
`gemmologists’ practice a brilliant cut with a deep pavilion, as in figure 4, is called
`‘dark centre stone’ [11].
`.
`
`As we change the other parameters of the well-proportioned cut of Tolkowsky,
`
`less homogeneous distributions and fewer points are obtained (figure 5). A similar
`
`comparison of various cuts (see the table) reveals that only the ones considered good
`
`by the experts (Tolkowsky, Scandinavian and practical fine cut) have a homogeneous
`distribution of the far—field pattern (figure .6). It should be noted, though, that the
`scan angle of 60° is quite arbitrary and any angle of about the same magnitude could
`
`Lorge Table
`
`0 (JOE
`4' 0.905?
`O
`,W+ + + +
`:2-.054:
`obese
`
`2
`
`Figure 5. Computed reflection patterns for a brilliant cut diamond having a large table and a
`diamond with small crown angle. All other details as in figure 4.
`
`Comprehensive review of types of brilliant cut [12].
`
`Johnson Tolkowsky
`
`Scandinavian
`
`Crown height
`Pavilion depth
`Table diameter
`
`In per cent
`of girdle
`diameter
`
`Crown main
`facets
`Pavilion main
`facets
`
`_ Angle to
`plane of
`girdle (deg.)
`
`19-2
`40-0
`53-1
`
`41-1
`
`38-1
`
`162
`43-1
`53-0
`
`34-5
`
`14-6
`431
`57-5
`
`34-5
`
`40-75
`
`40-75
`
`Practical
`fine
`
`cut
`
`14-'4
`43-2
`56-0
`
`33-2
`
`40-3
`
`Ratio of crown height to
`pavilion depth
`
`_
`_
`1 . 2 66'
`
`_
`_
`l . 3 00
`
`_
`_
`1 . 413
`
`Page 7 of 11
`
`

`
`306
`
`A. Hardy et all.
`
`as well be selected. On the other hand, a much wider or a much narrower scan angle is
`less useful, since under common Observation practice the gem is tilted back and forth
`through moderate angles. In Tact, when the light distribution in a 120° scanning
`angle is checked, Tolkowsky's cut
`is no longer characterized by the most
`homogeneous pattern of far-field spots (figure 7). Thus, one may conclude that a
`correlation exists between practical stone cuts and the degree to which their far—field
`light distribution is homogeneous within a scanning angle of about 60°'f.
`It is interesting to check, in the light of this criterion, other cuts and proportions
`which are not in practical use at present. In figure 8 we present the far-field pattern of
`a very deep cut. Except for the pavilion angle, all other parameters are exactly the
`same as in figure 4. Although the pattern is not as homogeneous as Toll<owsky's cut
`and the number of spots is obviously smaller, it still represents a cut which is better
`than the deep one of figure 4, even though the pavilion angle here is smalleri. Many
`
`-v
`
`0.0021.
`swag...
`
`Johnsons Cut
`
`,12KVZ§s\
`
`Scandinavian Cut
`
`SOUL‘:+
`
`you
`ooay
`coon? 5305’ aocizz
`
`Practical Fine Cut
`
`*1
`
`0.
`
`CV38
`+
`‘NW
`
`{f O
`
`0.00 3"1
`OO1'O + +
`/
`once?
`
`Po rker Gut
`
`Figure (1. Computed reflection patterns of various diamond cuts (see the table for the
`relevant parameters). All other details as in figure 4.
`
`1* By the term ‘homogeneous distribution’ we mean an even spread of spots of light which
`are of about the same intensity.
`II: should be pointed out that a homogeneous distribution is by no means the only
`criterion according to which the‘ quality of gem cut should be considered [9]. Nevertheless,
`when two distributions are so markedly difierent, a conclusion from this criterion alone is
`quite reasonable.
`
`Page 8 of 11
`
`

`
`A ray tracing study of gem quality
`
`807
`
`of the additional spots, in comparison with the deep cut of figure 4, can be attributed
`to an increase in the relative importance of the lower girdle facets over the pavilion
`facets (see figure 2). As a result, many rays emerge from the meridional plane of the
`brilliant cut. Thus, the use of secondary facets in the design of new cuts may provide
`additional ways to control the optical properties of a gem. This may include a
`brilliant cut with an odd number of facets, as first proposed by Elbe [13]. Note,
`however, that the two-dimensional models are no longer suitable for tackling the
`problem properly and ‘three-dimensional ones are necessary. Of these, the computer
`ray tracing method [8, 9] was the only one to be used so far.
`
`Shallow Cut
`
`Roe?
`‘ .
`T
`: onfianggijogfz A01:02:
`‘\
`4.
`0
`ll \
`cl auulzfi um£\\'-IN
`DDGIU
`D0050
`.
`
`'
`
`0
`0
`
`|.
`v 5
`.
`4.
`0
`;
`
`I
`
`00
`
`0.0032
`00 I2
`1-
`If
`u
`/
`J0092]
`D.0lflT fauna.
`
`Figure 7. Reflection patterns (right column) within 120° scan angle, for the diamond cuts of
`the left column.
`
`Very Deep Cut
`
`QQOEI DDBI4
`ems:
`*
`
`Figure 8. The reflection pattern of a diamond with a very deep pavilion. For other details see
`figure 4-.
`
`To study the wavelength dependence of the light distribution, computer
`programs with different indices were run. It was found that a slight index change
`modifies the spots’ locations and contributes to the effect known as ‘fire’. Large index
`variations may significantly change thepattern. For instance, when various cuts in
`glass (rt =1'S) were checked, the conventional Tolkowsky’s cut no longer produced
`the most homogeneous distribution and a shallow cut turned out to be better (figure
`9) in accordance with the previously mentio'ned'correlation with cuts considered best
`by the diamond trade.
`
`Page 9 of 11
`
`

`
`A. Hard_1' et al.
`
`well Proportioned Cut
`
`Figure 9. Reflection pattern of an eightfold symmetric brilliant cut glass (n=l-5) poly-
`hedron. All other details as in figure 4-.
`
`4. Conclusion
`
`In the previous section some computed reflection patterns of various gem cuts
`were shown. The spots obtained are records of the angles and intensities of emergent
`rays of light at the Fourier plane of a lens (far field). A correlation was sought between
`those patterns and possible good stone cuts. Since the sensation of beauty is
`subjective, the only reasonable way to define a criterion for a good stone is to find
`what is common to all of the gem cuts considered good by experts. Such an app roach
`led us to the requirement that the number of spots (of about the same intensity) in the
`far-field pattern of the scattered light should be as large as possible and homo-
`geneously distributed, all within a cone of about 60°. A much larger scanning cone,
`say of 120°, or a much smaller one will not do. The criterion found isjust one helpful
`tool for selecting either already existing good cuts or ones never tried before. Since
`the sensation of beauty is a complicated and involved matter and among other factors
`depends on light—eyt~mind interaction, it is not surprising to find that one single
`criterion is not sufficient. Such an approach was used by Dodson [9] who
`mathematically defined the previously vague concepts of brilliance, sparklincss and
`fire. His definitions, however, are integral ones in order to minimize the number of
`decisions one must take. This approach is quite useful, but with the help of such a
`powerful tool as a computer at hand, more parameters may and should be taken into
`account. To complicate matters, one should keep in mind that in practice an all
`important and economical requisite is that the stoneshould be cut to yield as much
`weight as possible. With this in mind, there is the possibility that cuts ideal according
`to all other criteria are no longer the most profitable. To avoid losses and maximize
`income, an operations—research kind of-decision-making should be considered which
`involves a compromise between different, sometimes contradicting, requirements.
`Such, for instance, could lead one to the choice of the very deep cut (figure 8) which is
`superior to the deep cut (figure 4) but still less attractive than 'I‘olkowsky’s cut. Here
`and under other complicated, but still realistic conditions, a spot diagram of
`scattered light rather thanjust one single number, may give a better estimate of what
`designs might appear more appealing to the human mind—eye system.
`In the study described above we only concentrated on the far field pattern. This,
`however, is the first step in the study of gemstones. As already mentioned in the text,
`
`Page 10 of 11
`
`

`
`A ray tracing study of gem quality
`
`809
`
`the program also enables one to study the light emerging from each facet separately.
`Thus dark and light regions on the diamond’s surface can properly be accounted for.
`Another important concept, which can be investigated through a ray tracing
`program,
`is ‘scintillation’. This is essentially the sensitivity of the pattern of
`reflections to small rotations of the gem. The higher this sensitivity is the greater is its
`effect. Therefore, one would like to maximize the relative motion of the scattered
`
`rays, with respect to a fixed reference point, clue to a dificerential rotation of the
`gemstone. With a diffused, spherically and evenly incident source light, the concept
`of scintillation is closely related to Dodson’s ‘sparkliness’ [9]. This, however, is not
`always the case. Very often it may happen that brighter sources of light are located
`somewhere not too close to the diamond thus‘ virtually producing a new light
`distribution with each diamond orientation. Irrthis case the autocorrelation of one
`
`single light distribution, i.e. ‘sparkliness’ is not directly related to ‘scintillation’.
`The concept of fire represents the sensitivity of the reflection pattern to small
`changes in wavelength. For white Iight sources this concept expresses the dispersion
`power of the gem due to the wavelength dependence of the refractive index.
`Obviously, for a non-white source the detailed spectral structure of the light plays an
`important role in the resulting colour effect of the gem. This effect can conveniently
`be studied through computer codes like the one ‘used for our study in this paper.
`
`Une technique de calcul de marche de rayons tridimensionnelle par ordinateur est utilisée
`pour obtenir le diagrarnme de diffusion de la lumiere incidente perpendiculairement sur la
`face d’une pierre précieuse. On rnontre qu’il existe une correlation entre les tailles de pierres
`précieuses considérées comme lcs me-illeures dans la profession des diamantaires et le
`diagramrne de réflexion s l’intérieur d’un angle de cone d’environ 60°.
`
`Eine computerisierte dreidimensionale Strahldurchrechentechnik wird zur Ermittlung
`cler Streulichtverteilung von senkrecht auf Edelsteine einfallendern Licht benutzt. Es wird
`ferner gezeigt, dafi eine Korrelation besteht zwischen den vom Diamantengewerbe als beste
`eingestuften Edclsteinschlitfen und dern inncrhalb eines Kegels von 60° reflektierten
`Lichtrnuster.
`
`References
`
`[1] BURTON, E., 1970, Diamonds (Radnom: Chilton Book Co.).
`[2] ToLI{owsKY, M., 1919, Diamond Design (London: E. & F. N. Spon).
`[3] JoHNsoN, A., 1926, S1721‘. preuss. Akad. Wise, 19, 322.
`[4] EULITZ, W. R., 1968, Gems Gemoll, 22, 263.
`[5] Rose!-I, S., 1926, Dr. Gaidsckmiedeztg, Nos. 5, 7, 9.
`.[6] ELBE, M., 1972, Z. dt. Gemmcl. G'es.,21, 189; 1973, Ib£a'.,22, 1; 1975; U.S. Patent No.
`3,858,979.
`[7] Baa-Isaac, C., FREI, E. H., and SHTRIKMAN, 8., 1970, Israel Patent No. 434-65, U.S.
`Patent No. 3,947,120.
`‘
`[3] STERN, N., 1975, M.Sc. Thesis, Weizmann Institute of Science, Rehovot, Israel. HARDY,
`A., SHTRIKMAN, 3., and STERN, N., 1977, Bull. Israelphys. Soc., 23, 98.
`[9] DODSON, J. S., 1978, Optics Acts, 25, 681, 693, 701.
`[10] BORN, M., and WOI.F, E., 1959, Principles of Optics (New York: Pergamon Press),
`pp.40—46.
`[11] HARDING, B. L., 1975, Gems Gemo£., 15, 78.
`[12] PAGEL-THEISEN, V., 1973, Handbook of Diamond Grading, fourth edition (Antwerp:
`R. Rubin & Sons).
`[13] ELBE, M., 1971, Z. Dr. Gemmoi. Ges., 20, 57.
`
`Page 11 of 11