Optical Properties
The most important optical properties of a material are its internal and external transmittances, surface reflectances, and refractive indices. The formulas that connect these variables in the on-axis case are presented below. |
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Transmission |
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External transmittance is the single-pass irradiance transmittance of
an optical element. Internal transmittance is the single-pass irradiance
transmittance in the absence of any surface reflection losses
(i.e., transmittance of the material). External transmittance is of
paramount importance when selecting optics for an image-forming lens
system because external transmittance neglects multiple reflections
between lens surfaces. Transmittance measured with an integrating sphere
will be slightly higher. Let Te
denote the desired external irradiance transmittance,
Ti the corresponding internal
transmittance, t1 the
single-pass transmittance of the first surface, and
t2 the single-pass
transmittance of the second surface: |
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where e is the base of the natural system
of logarithms, μ is the absorption
coefficient of the lens material, and tc
is the lens center thickness. This allows for the
possibility that the lens surfaces might have unequal transmittances
(for example, one is coated and the other is not). Assuming that both
surfaces are uncoated, |
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where |
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is the single-surface single-pass irradiance reflectance at normal incidence
as given by the Fresnel formula. The refractive index
n must be known or calculated from the
material dispersion formula given in the next section. These results are
monochromatic. Both μ and n are functions
of wavelength. To calculate either Ti or the Te for a lens at any wavelength of interest, first find the value of absorption coefficient μ from the equation given below. Typically, internal transmittance Ti is tabulated as a function of wavelength for two distinct thicknesses Tc1 and Tc2, and μ must be found from these. Thus |
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where the bar denotes averaging. In portions of the spectrum where
absorption is strong, a value for Ti
is typically given only for the lesser thickness.
Then |
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When it is necessary to find transmittance at wavelengths other than
those for which Ti is tabulated,
use linear interpolation. The on-axis Te value is normally the most useful, but some applications require that transmittance be known along other ray paths, or that it be averaged over the entire lens surface. The method outlined above is easily extended to encompass such cases. Values of t1 and t2 must be found from complete Fresnel formulas for arbitrary angles of incidence. The angles of incidence will be different at the two surfaces; therefore, t1 and t2 will generally be unequal. Distance tc, which becomes the surface-to-surface distance along a particular ray, must be determined by ray tracing. It is necessary to account separately for the s- and p-planes of polarization, and it is usually sufficient to average results for both planes at the end of the calculation. |
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Refractive Index and Dispersion |
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The Schott Optical Glass catalog offers nearly 300 different optical
glasses. For lens designers, the most important difference among these
glasses is the index of refraction and dispersion (rate of change of
index with wavelength). Typically, an optical glass specified by its
index of refraction at a wavelength in the middle the visible spectrum,
usually 587.56 nm (the helium d-line), and by (the Abbé vd-value, defined
to be |
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The designations F and C stand for 486.1 nm and 656.3 nm, respectively.
Here, vd shows how the index
of refraction varies with wavelength. The smaller
vd is, the faster the rate of
change is. Glasses are roughly divided into two categories: crowns and
flints. Crown glasses are (those with nd < 1.60 and vd
> 55, or nd > 1.60 and
vd > 50. The others are
flint glasses. The refractive index of glass from 365 to 2300 nm can
be calculated by using the following formula: |
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Here l, the wavelength, must be in micrometers,
and the constants B1 through
C3 are given by the glass
manufacturer. These constants are provided for most of the individual
glasses in this catalog. Values for other glasses can be obtained from the
manufacturer's literature. This equation yields an index value that is
accurate to better than 1 x 10-5 over the entire transmission range, and
even less in the visible spectrum. |
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Other Optical Characteristics |
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Homogeneity within Melt |
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Homogeneity within melt is the amount of refractive index variation
within the manufactured glass blank. Inhomogeneity of refractive index
can result in transmitted wavefront distortion. The maximum value for
homogeneity within melt for all Schott optical glasses used in Melles
Griot catalog components is 1 x 10-4. |
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Striae Grade |
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Striae are thread-like inclusions within an optical glass. Striae grades
are specified in U.S. military specification MIL-G 174B. All Melles
Griot catalog components that utilize Schott optical glass are specified
to have striae that conform to MIL-G 174B grade A. Grade A means that no
visible striae, streaks, or cords are present in the glass. |
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Stress Birefringence |
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Mechanical stress in optical glass leads to birefringence (anisotropy
in index of refraction) which can impair the optical performance of a
finished component. Optical glass is annealed (heated and cooled) to
remove any residual stress left over from the original manufacturing
process. Schott Glass defines fine annealed glass to have a maximum of
12 nm/cm of residual stress birefringence for blanks of up to 800 mm in
diameter and 100 mm in thickness. |
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| Optics Guide Copyright 2002 Melles Griot Inc. |