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4.7. Field curvature   ▐    4.8.1. Secondary spectrum and spherochromatism
 

4.8. Chromatic aberrations

Chromatic aberration, as the name implies (khroma=color in Greek) is the phenomenon of white light wavefront splitting into its spectral components' (i.e. individual wavelengths) wavefronts, as light encounters media of different optical density. It is caused by different wavelengths moving at different speeds through media denser than air. Denser media slow down all wavelengths, but shorter wavelengths more than longer wavelengths, in inverse proportion to the media index of refraction (the default value for air is 1, with the actual value - 1.00027784 for 550nm wavelength - being negligibly higher than for vacuum).


FIGURE 62
: Refractive index for the green e-line (546nm, or 0.546μ Fraunhofer spectral line) of the visual spectrum is for most optical glasses in the 1.4-1.9 range; plot to the left shows variation of refractive index with wavelength for several optical glasses or materials, and water. The higher refractive index, the stronger refraction, i.e. power of a refractive surface, or a lens.

As the plot illustrates, refractive index declines with the increase in wavelength, more rapidly going from short toward longer wavelengths, nearly flattening out in the infrared (index change is gradual in 0.2-4μ range, approximately; outside of it becomes much more volatile, particularly for long waves, about 10 microns and longer).

Refractive index also generally increases with glass density; it is lower for crown and higher for flint glasses. In the latter, index of refraction increases with their lead content.

Some other materials used for lens fabrication, such as calcium fluorite (form of crystal) and plastics like acrylic or polycarbonate, similarly to glass, refract more strongly shorter wavelengths, while gradually flattening out from the mid visual range (approximately) toward red and infrared..
 

As illustrated on Fig. 4, points of wavefront that simultaneously moves through media of different optical densities, generate wavelength-specific phase difference, due to different wavelengths now moving at different speeds through the incident and refractive media. This causes change in a form of the wavefront (in terms of rays, it results in refraction).

 Obviously, chromatic split of the wavefront will take place even if the shape of optical surface coincides with the shape of wavefront, for instance, when flat wavefront passes through an optical window at zero inclination angle. However, this effect is negligibly small; in BK7 glass, the green e-line is only 0.241% faster than blue F-line, and 0.29% slower than red C-line. In, say, 5mm thick window, axial separation between blue and red wavefront at the rear surface would be 0.0266mm.

Much more important is the effect of change in the wavefront form when the shapes of the wavefront and optical surface don't coincide - a common scenario with telescope lens objectives, made to change flat incoming wavefronts into spherical.  Here, the in-glass light speed varying with the wavelength causes shorter (faster) wavelengths to form more strongly curved wavefront - or, in terms of rays, refract more strongly - and focus closer than longer wavelengths (FIG. 63). The aberration's ray geometry is governed by the law of refraction, stating that the sines of the incident (α) and refracted (α') ray angle to the surface normal relate in inverse proportion to the refractive indici of the incident and refractive media, or sinα'/sinα = n/n', with n and n' being the incident and refractive media refractive index, respectively.

FIGURE 63: The three forms of chromatic aberration are: (1) chromatic defocus, (2) lateral color and (3) spherochromatism. Illustrations above show how the the first two form in a lens, grossly exaggerated. LEFT: white-light wavefront W splits into its component wavelengths wavefronts, as shorter wavelengths lag behind in a denser media of the refractive n', forming more strongly curved wavefronts with shorter foci. This form of chromatism is called primary color, or primary spectrum: the longitudinal variation of color foci produced by a single lens, where the focus falls farther as the wavelength increases. The path of refracted rays (dotted lines) is orthogonal with respect to the transforming wavefronts. Note that the axial separation of the wavefronts themselves is exaggerated; typically, it is negligible in comparison to the effect of wavefront radii variation (i.e. the extent of their focal length variation), also grossly exaggerated in the illustration. In a doublet achromat, primary color is corrected by bringing two widely separated wavelengths - usually red C and blue F lines - to a common focus, but other wavelengths still focus differently. This longitudinal focus disparity for other wavelengths vs. common focus of two widely separated wavelengths is called secondary spectrum (usually measured by the displacement of e- or d-line focus). Strictly talking, only a single wavelength can come out of a lens objective with perfectly spherical wavefront; the rest of them are affected by spherical aberration. Variation of spherical aberration with the wavelength is called chromatic spherical aberration, or spherochromatism (not shown above). RIGHT:  The white-light chief ray (the one orthogonal to the center of the incident wavefront at an angle to the optical axis), splits laterally into the component wavelengths chief rays, setting the stage for lateral chromatism; unlike primary and secondary color, and spherochromatism, which are axial aberrations (meaning that they affect the entire field), lateral chromatism is off-axis aberration, caused by variation of refraction power with the wavelength for off axis point. With the stop at the surface, it passes through the center of the lens, which acts as a plane-parallel plate, having the chief rays of different wavelengths exit slightly separated, but at nearly identical angle. Consequently, lateral chromatism approaches zero, unless the lens is exceedingly thick. With the aperture stop displaced from the lens, the chief ray passes through a portion of lens with significant optical power, resulting in chief rays of different wavelengths exiting the lens ad different angles, thus focusing at different heights in the image space. Since lateral color and secondary spectrum have different origins and magnitude, correcting one doesn't necessarily cancel the other.

Independently of the longitudinal chromatic error - either primary or secondary spectrum, which result from defocus error relative of other wavelengths relative to a specific wavelength - every refractive objective also generates spherochromatism, as spherical aberration can be optimally corrected only for a narrow range of wavelengths. Depending on the specifics of a lens, or lens group, spherochromatism ranges from entirely negligible to a serious detriment to optical quality. Both of these chromatic aberrations are axial aberration, i.e. affect the entire image field. The third main type of chromatic aberration is lateral chromatism, causing abaxial point images to be formed at different heights for different wavelengths, thus smearing the spectral energy radially. When present, it is zero on axis (assuming properly aligned lenses), increasing with the field angle.

Longitudinal chromatic aberration

This form of chromatic aberration is traditionally in the focus of amateur astronomers, because it is the dominant form of chromatic aberration in achromats - a type of telescope that remained popular among amateurs long time after it was abandoned by professional astronomy.

The cause of longitudinal chromatism is variation of glass' refractive index with the wavelength, and with it the inverse to it variation in lens' focal length (FIG. 64).


FIGURE 64
: General scheme of the change in glass' refractive index with the wavelength, and accompanying change in lens' focal length. As the plots show, refractive index decreases with the wavelength, while the resulting focal length, expectedly, follows the opposite dynamics. The pace of change in the refractive index also decreases as the wavelength increases, and so does the pace of the resulting change in focal length. While lens focal length for any given wavelength is determined by the glass' refractive index for that particular wavelength, the change in focal length as a function of change in the wavelength and its corresponding index depends on the dynamics of index change with the wavelength - or glass dispersion. It varies from one glass type to another. As it will be addressed in more detail ahead, it is this variation in dispersion from one glass type to another that made possible overcoming the crippling chromatism of a single lens.
Since both, refractive index and dispersion are determinants of the chromatic optical properties of a material, standard glass diagram groups glasses with respect to their refractive index and dispersion.

Longitudinal chromatism of a single lens - known as primary spectrum - is overwhelming: it smears the wavelengths into a rainbow of colors, due to shorter wavelengths, for which glass has higher refractive indici, refracting more strongly and focusing closer than longer ones. In terms of lens power, shorter wavelengths converge more strongly in a positive lens, and diverge more strongly in a negative lens. Hence red focuses farther away from blue in both, positive or negative lens (with the focus in the latter being imaginary, determined by extending refracted rays in the direction opposite to their travel).

For object at infinity, longitudinal chromatism of a single lens, known as primary chromatism, defined as the separation between red C-line and blue F-line foci, is proportional to 1/V, V being the glass dispersive constant, or Abbe number. Specifically, this means that a lens of the focal length ƒ will have longitudinal chromatism, as defined above, equal to ƒ/V. The blue focus, which focuses shorter, is somewhat closer to the e-line focus than the longer red focus, at approximately 45% of the F-C separation.

General form of the Abbe number is:

with n being the glass refractive index, and ι the index differential for the chosen range. The usual choice for n is either e (λ=0.5461μ) green, or d (λ=0.5876μ) yellow line, and for ι the index differential between the blue F-line (λ=0.4861μ) and red C-line (λ=.6563μ), nF-nC. Taking e line and BK7 glass (ne=1.518722, nF=1.522376 and nC=1.514322), gives the corresponding Abbe number as:

Thus, the lower Abbe number, the higher glass dispersive power, and vice versa.

Lens' lateral color is near-zero with the stop at the surface; for displaced stop, it increases with stop displacement, in proportion to the field angle, depending on lens' dispersive and optical powers, as well as its thickness). However, its longitudinal chromatism is independent of stop location. A single BK7 lens of the focal length ƒ=1000mm, would have the axial F-C separation of over 15mm. In other words, it would be all but useless for observing of pretty much anything but the chromatism itself.

According to raytrace, a single 100mm diameter crown lens needs to be about ƒ/340 in order to achieve 0.80 polychromatic Strehl in the visual range (for photopic eye sensitivity). The required ƒ-ratio scales with the aperture diameter, thus 50mm lens needs to be "only" ƒ/170. Obviously, longitudinal chromatism for given glass changes in proportion to the focal length. However, the effective error, corresponding to the polychromatic Strehl, changes at a somewhat slower rate. For instance, an ƒ/170 100mm lens will have 0.55 polychromatic Strehl, not ~0.4 that would result from the λ/2 wave P-V of spherical aberration equivalent (doubled λ/4 wave equivalent at ƒ/320). Similarly, at ƒ/640, its polychromatic Strehl is about 0.93, corresponding to λ/7 wave P-V of spherical aberration, not nearly 0.95 Strehl that would result from λ/8 wave.

The above numbers imply that the defocus tolerance at F- and C-line for 0.80 poly-Strehl level in a single lens is just over λ/2 P-V.

In order to minimize longitudinal chromatic defocus in a lens objective, it has to be made of two or more glass elements of different optical properties. Such compounded refracting objectives usually consist of two or three lens elements. Depending on the level of correction of chromatism, most of them belong to one of the two main groups of lens objectives, achromats and apochromats. In the former, primary spectrum of a single lens is corrected by bringing two wavelengths near the opposite ends of the visual spectrum (usually blue F- and red C-line) to a common focus. However, other wavelengths still deviate from this focus, resulting in so called secondary spectrum. Using glass of abnormal dispersion (i.e. significantly different from that in common-type glasses) for one of the elements, allows for the correction of secondary spectrum, with the only potentially significant form of chromatism remaining being spherochromatism.

Not every abnormal glass has dispersion differential strong enough to effectively eliminate secondary spectrum. In that case, it is only partly reduced; such objectives - along with those having secondary spectrum cancelled, but still retaining strong spherochromatism - belong to somewhat vaguely defined class of semi-apochromats.

Optical glasses

Obviously, refraction is a more complex phenomenon than reflection. While the reflected angle is always identical to the incident angle, relative to the normal to a surface, angle of refraction varies with the optical properties of a refractive medium. In general, denser glass slows light propagation more, hence it refracts light more strongly and, consequently, has higher refractive index.

While the glass' index of refraction and dispersion, as mentioned, are its two most important optical properties, it is also of significance how much of the light incident upon glass surface is reflected back (that applies to both, air-to-glass and glass-to-air surface), and how much of the light that enters glass medium actually makes it to its rear surface (FIG. 65

FIGURE 65: Two important properties of the optical glass are reflectance and transmittance. Both are dependant on the glass' refractive index, but in a different way.

Plot at left illustrates these dependencies in terms of the relative index of refraction n*, defined as a ratio of the index of incidence vs. transmittance medium, or n*=nt/ni. When the incidence medium is air (ni=1), n* equals the refractive index of glass.

Most optical glasses fall in the 1.4<n<1.8 refraction index range, with the corresponding air-to-glass reflectance (R) and transmittance (T) in the 3-8% and 8-3% range, respectively. For glass-to-glass surface, n* values are most often in the proximity of 1, with near-zero reflectance and nearly total transmittance.

Obviously, the two curves are symmetrical, i.e. complementary, and either value can be expressed in terms of the other one: as T=1-R (as a ratio number; 100-R for T and R in percents) and R=1-T.

As mentioned, the parameter specifying the effect of an optical surface, or material, on propagation of light is called reflective and refractive index, for reflective and refractive media, respectively. In the case of reflection, there is no change in speed of light, only direction, which is generally opposite to that of incident light. Consequently, for surface in air (refractive index n=1), the reflective index is n'=-1.

However, after passing refractive surface and entering different medium, speed of light in it changes, and that also results in the change of light direction whenever shape of the incident wavefront does not coincide with the surface shape, as FIG. 4 illustrates. Since general direction of light propagation does not change, glass refractive index has the same sign as the media of incidence (usually air, with refractive index n=1, or n=-1, depending on the direction), with its nominal value, as mentioned, given as an inverse of the speed of light in glass vs. that in vacuum (which is exactly 1, with the refractive index of air, 1.00027784 for 550nm wavelength, being negligibly higher).

The slow-down of light propagation inside glass varies with the wavelength; in general, the shorter wavelength, the more it is slowed down. The greater slow-down, the stronger refraction. This variability of refraction with the wavelength is the origin of chromatic aberration. The relative strength of refraction for different wavelengths - or glass dispersion - as mentioned above, is expressed with Abbe number. Due to their different physical properties, the dispersive properties vary somewhat from one glass to another. This fortunate phenomenon, expressed with a parameter called relative partial dispersion of glass, made possible to correct one major form of chromatic aberration, secondary spectrum.

Summing it up, main properties of optical glass are expressed with its index of refraction, its dispersion (Abbe) number and its relative partial dispersion. Following graph shows optical glasses of two major manufacturers grouped according to their refractive index and dispersion.



FIGURE 66
: ne-Vd glass diagram for Schott and Ohara glasses.

As the Shott diagram shows, the traditional crown and flint denominations for optical glasses indicates refractive index n1.6 and Abbe number V55 for the former, and n1.6, V55 for the latter. That classification is, however, very general; there are "transitional" glasses between crown and flint, and some modern crowns have index of refraction significantly higher than 1.6. The main ingredient of crown glasses are alkali-lime silicates, with their optical properties varying with the secondary ingredient, the common being potassium oxide (crown), boric oxide (borosilicate crown), barium oxide, zinc oxide, phosphorus pentoxide, fluorite and lanthanum oxide. Flint glasses have high lead content, which is gradually being lowered or replaced by less toxic substances.

Common glass, usually called soda-lime float, or plate glass varies somewhat in its optical properties. Its e-line index of refraction is about 1.52 and its Abbe number about 60.

In general, the denser glass, the lower its Abbe number and the higher its refractive index, usually given for e- or d- Fraunhofer line, as ne and nd, respectively. Since all the wavelengths participate in the formation of a telescopic image, the presence of refractive elements makes it necessary to determine its effect within entire spectral range of interest. For the visual spectrum, it encompasses wavelengths from ~0.4 micron (μ) to ~0.7μ.

The usual way of assessing and presenting the effects of refraction is by selecting up to a several selected spectral lines as representatives of the specific portions of the wavelength range. The most commonly used spectral lines notation is that established by Fraunhofer. Some of the Fraunhofer's spectral lines routinely used in analyzing optical properties of telescopes with refracting elements are listed below.
 

Wavelength

Fraunhofer line

Emitting element

Color

microns (μ)

nanometers (nm)

Εngstrφm (Ε)

0.405

404.7

4046.6

h

mercury (Hg)

violet

 

0.436

435.8

4358.4

g

mercury (Hg)

violet

0.480

480.0

4799.9

F'

cadmium (Cd)

blue

0.486

486.1

4861.3

F

hydrogen (H)

blue

0.546

546.1

5460.7

e

mercury (Hg)

green

0.588

587.6

5875.6

d

helium (He)

yellow

0.589

589.3

5893.0

D

sodium (Na)

yellow

0.644

643.8

6438.5

C'

cadmium (Cd)

red

0.656

656.3

6562.7

C

hydrogen (H)

red

0.707

706.5

7065.2

r

helium (He)

red


4.7. Field curvature   ▐    4.8.1. Secondary spectrum and spherochromatism
 

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