16 |
The Work of the Lens Designer |
The advantages of plastic lenses are:
1.Ease and economy of manufacture in large quantities.
2.Low cost of the raw material.
3.The ability to mold the mount around the lens in one operation.
4.Lens thicknesses and airspaces are easier to maintain.
5.Aspheric surfaces can be molded as easily as spheres.
6.A dye can be incorporated in the raw material if desired.
The disadvantages are:
1.The small variety and low refractive index of available plastics.
2.The softness of the completed lenses.
3.The high thermal expansion (eight times that of glass).
4.The high temperature coefficient of refractive index (120 times that of glass).
5.Plane surfaces do not mold well.
6.The difficulty of making a small number of lenses because of mold cost.
7.Plastics easily acquire high static charges, which pick up dust.
8.Plastic lenses cannot be cemented41 and can be coated only with some difficulty.42
In spite of these issues, plastic lenses have proved to be remarkably satisfactory in many applications, including low-cost cameras, and as manufacturing and materials technologies advance, so will the variety of applications. In some cases, glass and plastic lenses have been used together effectively in optical systems.
1.4 INTERPOLATION OF REFRACTIVE INDICES
If we ever need to know the refractive index of an optical material for a wavelength other than those given in the catalog or used in measurement, some form of interpolation must be used, generally involving an equation connecting n with l. A simple relation, which is remarkably accurate throughout the visible spectrum, is Cauchy’s formula43:
n ¼ A þ B=l2 þ C=l4
Indeed, the third term of this formula is often so small that when we plot n against 1/l2 we obtain a perfectly straight line from the red end of the visible almost down to the blue-violet. For many glasses the curve is so straight that a very large graph may be plotted, and intermediate values picked off to about one in the fourth decimal place.
To use this formula, and the similar one due to Conrady,44 namely,
n ¼ A þ B=l þ C=l7=2 |
(1-1) |
1.4 Interpolation of Refractive Indices |
17 |
It is necessary to set up three simultaneous equations for three known refractive indices and solve for the coefficients A, B, and C. In this way indices may be interpolated in the visible region to about one in the fifth decimal place.
Extrapolation is, however, not possible since the formulas break down beyond the red end of the spectrum.
Toward the end of the last century, several workers, including Sellmeier, Helmholtz, Ketteler, and Drude, tried to develop a precise relationship between refractive index and wavelength based on resonance concepts.45 The one most generally employed is
n2 ¼ A þ |
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þ |
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D |
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F |
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(1-2) |
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2 |
C |
2 |
l |
2 |
E |
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G |
2 |
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l |
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l |
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In this formula the refractive index becomes infinite when l is equal to C, E, G, and so on, so that these values of l represent asymptotes marking the centers of absorption bands. Between asymptotes the refractive index follows the curve indicated schematically in Figure 1.5.
For most glasses and other transparent uncolored media, two asymptotes are sufficient for interpolation purposes, one representing an ultraviolet absorption and the other an infrared absorption. The visible spectrum is then covered by values of l lying between the two absorption bands.
Expanding Eq. (1-2) by the binomial theorem, we obtain an approximate form of this equation, namely,
n2 ¼ al2 þ b þ c=l2 þ d=l4 þ . . .
in which the coefficient a controls the infrared indices (large l) while coefficients c, d, and so on, control the ultraviolet indices (small l). If the longer infrared is
n
1.0 |
log l |
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Visual |
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region |
UV absorption |
IR absorption |
Figure 1.5 Schematic relationship between the refractive index of a glass and the log of the wavelength.
18 |
The Work of the Lens Designer |
of importance in some particular application, then it is advisable to add one or more terms of the type el4 þ fl6, and so on.
Herzberger46 proposed a somewhat47 different formula, namely,
n ¼ A þ Bl2 |
þ |
C |
þ |
D |
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l2 l02 |
l2 l02 2 |
in which A, B, C, D are coefficients for any given glass, and l0 has a fixed value for all glasses. He found that a suitable value is given by l02 ¼ 0.035, or l0 ¼ 0.187. This takes care of the ultraviolet absorption, and the near infrared is covered by the Bl2 term. If the infrared is more important, another infrared term should be added.
In the first edition of this book, the then current Schott glass catalog contained a six-term expression used for smoothing the stated index data. It was
n2 ¼ A0 þ A1l2 þ A2=l2 þ A3=l4 þ A4=l6 þ A5=l8
which provided a very high degree of control in the blue and ultraviolet regions, but it is not valid much beyond 1 mm in the infrared. Since then, Schott has adopted the Sellmeier dispersion formula48 given by
s
nðlÞ ¼ |
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B1l2 |
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B2l2 |
B3l2 |
1 þ |
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þ l2 C3: |
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l2 C1 |
l2 C2 |
It should be noted that Schott now uses a nine-digit glass code where the first three digits represent the refractive index, the next three the Abbe value, and the final three the density of the glass. For example, the glass code for SF6 is 805254.518. Then nd ¼ 1:805 (note that 1.000 is added to the first three digits), vd ¼ 25:4 (second three digits are divided by 10), and the density is 5.18 (third three digits are divided by 100).
The Bausch and Lomb Company49 has used the following seven-term formula for its interpolation:
n |
2 |
¼ a þ bl |
2 |
þ cl |
4 |
þ |
d |
þ |
el2 |
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l2 |
l2 f þ gl2= l2 f |
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This is an awkward nonlinear type of relationship involving a considerable computing problem to determine the seven coefficients for any given type of glass.
1.4.1 Interpolation of Dispersion Values
When using the (D – d) method of achromatism (Section 5.9.1), it is necessary to know the Dn values of the various glasses for the particular spectral region that is being used. For achromatism in the visible, the Dn is usually taken
1.4 Interpolation of Refractive Indices |
19 |
to be (nF – nC), but for any other spectral region a different value of Dn must be used. Indeed, a change in the relative values of Dn is really the only factor that determines the spectral region for the achromatism.
To calculate Dn we must differentiate the (n, l) interpolation formula. This gives us the value of dn/dl, which is the slope of the (n, l) curve at any particular wavelength. The desired value of Dn is then found by multiplying (dn/dl) by a suitable value of Dl. Actually, the particular choice of Dl is unimportant since we shall be working toward a zero value of S (D – d) Dn, but if we are expecting to compare a residual of S (D – d) Dn with some established tolerance, it is necessary to adopt a value of Dl that will yield a Dn having approximately the same magnitude as the (nF – nC) of the glass.
As an example, suppose we are using Conrady’s interpolation formula, and we wish to achromatize a lens about some given spectral line. Then by differentiating Eq. (1-1), we get
dn |
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(1-3) |
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2 l9=2 |
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This formula contains the b and c coefficients of the particular glass being used, and also the wavelength l at which we wish to achromatize, say, the mercury g line.
Suppose we are planning to use Schott’s SK-6 and SF-9 types. Solving Eq. (1-1) for two known wavelengths, we find
Glass |
b |
c |
dn/dl at the g line |
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SK-6 |
0.0124527 |
0.000520237 |
–0.142035 |
SF-9 |
0.0173841 |
0.001254220 |
–0.275885 |
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For wavelength 0.4358 mm, we find for these two glasses that Dn ¼ 0.010369 and 0.020140, respectively, using the arbitrary value of Dl ¼ –0.073. These values should be compared with the ordinary Dn ¼ (nF – nC) values, which for these glasses are 0.01088 and 0.01945 respectively. It is seen that the flint dispersion has increased relative to the crown dispersion, which is characteristic of the blue end of the spectrum.
1.4.2 Temperature Coefficient of Refractive Index
If the ambient temperature in which the lens is to be used is liable to vary greatly, we must consider the resulting change in the refractive indices of the materials used. For glasses this usually presents no problem since the