Apochromatic Optical Design

Apochromatic vs. Conventional Periscope

Conventional periscope optical design has traditionally employed common achromat  relay and objective lenses to correct color aberrations. However, a very large amount of residual color aberration known as “secondary spectrum” still remains. Further color correction is needed  for sensors which cover a broader spectral band than the human eye. These include photographic film and electronic imaging arrays. “Apochromatic” design largely corrects the secondary spectrum and provides the broader band color correction needed for photographic and electronic sensor applications.

The following presentation analyzes the source of secondary spectrum, gives examples,  and provides the means for its  correction using special glasses. Considerations for the  use of special glasses are given which include design, manufacturing and operation in the field. Periscope image quality for apochromatic vs. ordinary achromat lenses is compared.

Secondary Spectrum The secondary spectrum of an achromat doublet based on ordinary crown and flint  optical glasses, Schott K7 and F2, is illustrated below.

Secondary Spectrum Plot The focal length chosen for this example is 100 mm. Color correction  is the traditional “C-F” correction. i.e., red-blue, or specifically, 656 and 486 nm wavelengths. The residual variation in the back focal length, Δf, measured from the green e-line at 545 nm is the secondary  spectrum as defined here. It amounts to about 0.050 mm in this example.

Secondary Spectrum Source

Secondary spectrum results from the dispersion properties of optical glasses. Of particular interest is the “relative partial dispersion” property of the glass. It is discussed here with reference to the diagram below. The definition is commonly found in glass catalogs, such as the referenced glass catalog excerpt.

Prism Dispersion

The relative partial dispersion of a glass over the wavelength range  λe to λF,  PF,e , is the ratio of the dispersion by a prism  over the  F-e  band to the dispersion over the F-C band:

                         P_{F,e}=\frac{\delta _{F}-\delta _{e}}{\delta _{F}-\delta _{C}}

The ray deviation angle, δ,  is a function of the prism angle, α,  and refractive index N:

                     \delta =(N-1)\alpha )

The expression for the relative partial dispersion for the e-F spectral band is therefore:

P_{F,e}=\frac{N_{F}-N_{e}}{N_{F}-N_{C}}

Prism Achromat Example Illustrating Secondary Spectrum

Secondary spectrum is presented here for an achromatic prism pair instead of the customary treatment given for an achromat lens doublet. This provides a slight simplification in analysis. The equivalency between the two will be evident.

Consider the achromatic prism  illustrated below, made up of crown and flint glasses.

Prism Achromat II

 

The crown glass element is designated by “a” and the flint element by “b”. The prism angles are αa and αb and  the prism refractive indexes, Na and Nb.

Deviation of the ray through the combination is Δ, with the subscripts referring to the C, F and e wavelengths. The C and F wavelengths exiting the prism pair are superimposed indicating  the existence of the achromatic condition. The separation of the e wavelength from the C and F wavelengths constitutes the secondary spectrum.

The deviations of the C and F rays through the combination of flint and crown prisms are:

\Delta _{C}=\left ( N_{a,C} -1\right )\alpha _{a}-\left ( N_{b,C} -1\right )\alpha _{b}

\Delta _{F}=\left ( N_{a,F} -1\right )\alpha _{a}-\left ( N_{b,F}-1 \right )\alpha _{b}

Setting the condition for achromatism, Δ= Δyields:

                                    \frac{\alpha _{a}}{\alpha _{b}}=\frac{N_{b,F}-N_{b,C}}{N_{a,F}-N_{a,C}}

That is, for achromatism the crown and flint prism angles are inversely proportional to their dispersions.

Secondary spectrum, ΔF-e = ΔC-e,  can now be expressed mathematically as the difference between the deviation of the e wavelength and the superimposed  C and F wavelengths:

\Delta _{F-e}=\Delta _{F}-\Delta _{e}

where                                          \Delta _{e}=(N_{a,e}-1)\alpha _{a}-\left ( N_{b,e} -1\right )\alpha _{b}

Substituting  the expressions for ΔF and Δe into the first equation and simplifying yields the expression for the secondary spectrum :

\Delta _{F-e}=\alpha _{b}\left (N_{b,F}-N_{b,C}\right )\left (\frac{N_{a,e}-N_{a,C}}{N_{a,F}-N_{a,C}} -\frac{N_{b,e}-N_{b,C}}{N_{b,F}-N_{b,C}} \right )

Letting Pa, e-C and Pb, e-C represent the relative partial dispersions for the “a” and “b” prism components over the e-C wavelength spectrum:

 \Delta _{F-e}=\alpha _{b}\left ( N_{b,F}-N_{b,C} \right )\left ( P_{a, e-C}-P_{b, e-C} \right )

As can be seen from the rightmost factor in the equation,  if the relative partial dispersions of the crown and flint elements are equal, the secondary spectrum will be zero; otherwise, it will have a value determined by the sum of the two relative partial dispersions multiplied by the product of the “b” prism angle and dispersion.

The deviation of the e ray through the prism pair can be rewritten as:

\Delta _{e}=\alpha _{b}\left ( N_{bF}-N_{bC} \right )\left ( \nu _{ae}-\nu _{be} \right )

The ratio of the secondary spectrum to the deviation through the prism pair is:

\frac{\Delta _{F-e}}{\Delta _{e}}=\frac{P_{a, F-e}-P_{b, F-e}}{\nu _{ae}-\nu _{be}}

Notes-

The above equations referencing the “b” prism angle and dispersion can alternately be expressed in terms of the “a” component prism angle and dispersion by employing the relationship between the two components for achromatism, derived above.

The secondary color for an achromat doublet lens is directly analogous to that for the achromat prism pair presented above. For a lens, the deviation of an incident ray is:

\delta =h\left ( N-1 \right )\left \left\left ( \frac{1}{R_{1}}-\frac{1}{R_{2}} \right )

where h is the height of the incident ray and R1 and R2 are the lens radii of curvature. Thus, the physical factor,   h (1/R1-1/R2)),  corresponds to the prism angle, α. The refractive index term is identical, and the equivalency between lens and prism is apparent.

“Normal” Glass vis-a-vis Secondary Spectrum

“Normal” glasses, which make up the large majority of optical glasses, have relative partial dispersions  that are linearly proportional to their V-numbers:

P_{x,y}=a_{_{x,y}}+b_{x,y}\nu_{d}}}

Therefore, in an achromat comprising “normal glasses”  a large separation in V-numbers leads to a large difference in the relative partial dispersions. Large separation in V-numbers in an achromat  is necessary to keep the dioptric powers of the elements at a minimum for control over spherical aberration. Thus, trying to achieve a match in the relative partial  dispersions for secondary spectrum reduction runs directly in opposition to the need for large V-number separation for the correction of spherical aberration.

An example of this linear relationship between the relative partial dispersion and V-number for the red-to-near-infrared wavelength region is shown in the chart, Fig. 1, taken from an earlier Schott glass catalog.

Secondary Spectrum Example for Normal Glasses

Consider an achromatic prism made up of Schott K7 crown and F2 flint glasses. Data from the Schott glass catalog for these “normal” glasses are shown in the table below.

Slide1

The secondary spectrum due to the mismatch in the relative partial dispersions between K7 and F2 for the d-F spectral band is:

\frac{\Delta _{F-d}}{\Delta _{d}}=\frac{P_{a, F-d}-P_{b, F-d}}{\nu _{ad}-\nu _{bd}}=\frac{0.4693-0.4567}{60.41-36.37}=0.0005

For all combinations of  “normal” crown and flint glasses, this ratio of secondary spectrum to the deviation of the ray through the prism pair is of nearly constant value, about 0.05%.

 “Special” Glasses: Correction of Secondary Spectrum:

Glass manufacturers have been aware of the need for glasses with partial dispersions which deviate from the  “normal” linear relationship and have developed “special” glasses for correcting secondary spectrum. The data is presented in tabular and graphic forms in glass catalogs, as shown in the  referenced Schott catalog excerpt..

Glasses with favorable matches in the  relative partial dispersions can be found by searching and examining the glasses in the glass manufacturer’s catalogs. Graphing techniques, computer data analysis and optical design software can facilitate this process.

The optical properties of two such glasses used in the apochromatic periscope example which follows are tabulated below:

Slide2

The secondary spectrum  in the d-F waveband for this combination of glasses is:

\frac{\Delta _{F-d}}{\Delta _{d}}=\frac{P_{a, F-d}-P_{b, F-d}}{\nu _{ad}-\nu _{bd}}=\frac{0.6946-0.6934}{81.61-54.01}=0.00004

This is very favorable match up in relative partial dispersions and leads to great improvement in secondary spectrum as will be seen below.

Special Glasses vs Normal Glasses:  Apochromat vs. Achromat

A doublet apochromat designed with special glasses greatly reduces the secondary spectrum color aberration  as compared to the doublet achromat presented at the outset of this presentation:

Apochromat vs Achromat BFL copy

Apochromatic Periscope

A periscope  modified to employ special glasses yields a dramatic improvement in secondary spectrum, particularly in wavelength areas beyond the human eye’s greatest sensitivity:

Periscope Focal Distance vs copy

The advantage of the apochromatic design is great for broadband sensors,

In terms of optical path difference, the wavefront error for wavelengths throughout the spectrum from blue (486 nm) to red (656 nm) is about  ±λ/4 for the apochromatic   persicope vs. 5λ for the periscope comprising orordinary achromats:

Pertiscope Wavefront Aberrations (Achromat vs Apochromat (merged)

Note the 4x scale difference in the above graphs.

In terms of the transverse ray aberrations at the eyepiece image plane:

Periscope HTan U Apochromjat vs Achromat 21 May 13 copy

Image Color Fringing

Color fringes on the image can sometimes be seen when viewing through a conventional periscope.The transverse ray aberration plot above  and the illustration below aid in the understanding of the “color fringing” phenomenon.  Secondary Spectrum Color Fringing

The difference in slopes of the aberration plots of the green and red-blue wavelengths indicates that their focal points lie in different planes along the axis. Green is in focus and appears to be at infinity for all positions of the eye across the exit pupil. On the other hand, red and blue light rays appear to be coming from a nearer location. This  causes an angular separation between the green and red-blue light rays emerging from the eyepiece. A color fringe on a dark object seen against a bright background or vice versa is thereby formed.

The magnitude of the color fringing can be calculated from the transverse ray aberration data. The angular difference between the  green and red-blue rays as they emerge from the eyepiece, Δθ, is the transverse ray abverration, Δy, divided by the eyepiece focal length, f . The angular spread then for  the transverse aberration, Δy = 0.125 mm, eyepiece focal length,  f = 71. 7mm is 1.74 milliradians or nearly 6 arc minutes, well in excess of the 1 arc minute resolution limit of the human eye.

Modulation Transfer Function

The periscope modulation transfer function (MTF) for a sensor having uniform spectral response from 400 to 700 nm, located at the periscope eyepiece image plane, is shown below for achromat and apochromat lens configurations.

Periscope MTFs Apochromats vs Achromats copyThe  MTF for eyepiece viewing with a 3.0 mm pupil is shown below for periscope  achromat and apochromat configurations. MTF wavelength weightings are based on the eye’s spectral sensitivity and D6500 scene illumination and are calculated at relatively small 10 nm intervals for maximum accuracy.

Periscope MTFs, Visual

The image quality is essentially perfect for the special glass apochromats configuration but still excellent for the ordinary glass achromats,  as well.

Further Image Quality Improvement via Optimal  Periscope Lens Layout 

Periscope secondary spectrum can sometimes be further reduced by a more optimal choice of the focal lengths of the objective and relay lenses. This can be accomplished without increasing the number of lenses or the diameters of the lenses.

The equation which expresses periscope secondary spectrum aberration, in terms of the lens configuration, is  derived in the document, “Secondary Spectrum Equations”:

\Delta _{\alpha }=\frac{K}{2F_{N}}\left \{\frac{f_{N}}{f_{R}}\left [ 1+\left ( N-1 \right )M \right ]+M\ \right \}             where,

                   K = 1/2200, the secondary spectrum constant for normal optical glasses

    FN = f-number of the Nth relay lens, i.e. the last relay lens

    fN = the focal length of the last relay lens

     fR = the focal length of all relay lenses but the last

     N = total number of relay lenses

     M = magnification of the telescope comprising the periscope first relay lens as the objective lens and the periscope objective lens as the eye lens.

M=\frac{f_{1}}{f_{_{o}}}

An example of the reduction in secondary spectrum achievable by optimal configuration of the objective lens and relay lens focal lengths is given in “Secondary Spectrum Equations”, page 4. It is shown that by increasing the objective lens focal length 2x while decreasing the focal length of the last relay lens by ½x so as to maintain the overall periscope focal length constant, the residual secondary spectrum is only 27% of the reference configuration.

Improved Apochromatic Periscope Aberrations copyPerhaps, equally important is the improvement in spherical aberration which follows when the relay lens f-numbers increase. Of particular importance is the variation of spherical aberration with wavelength, a principle limitation to diffraction limited image quality  in broad spectrum optical designs. The reduction in spherical aberration is evident in the graphs of the transverse ray aberrations shown at the left. This is indicated by the increased straightness of the plots at each wavelength.

The combined improvement  effects of reduced secondary spectrum and spherical aberration are summarized in the MTF plots below.MTF Improved Apochromatic Periscope copy

The MTF from 400-700 nm wavelength for the improved apochromatic periscope  shows a significant improvement over the previous design with less optimal layout of the objective lens and relays lenses.

As can be seen from the plot of the transverse ray aberrations, control of the deep blue wavelengths remains a challenge. However, if the wavelength region from 400 – 450 nm is omitted, either by optical filtering or specification of sensor spectral sensitivity,  then the MTF is very neqrly diffraction limitged as shown in the above plot.

Whether improvement as described above can be implemented in a given design can depend on other periscope design trade-offs. For further information and examples,  refer to  “Periscope Design Trade-offs”  a  preprint of an SPIE paper of the same title.

Special Glass Caveats

There can be certain negative side effects in the design, manufacture and field application of special glass optics.

In lens design, there is a natural tendency toward selection of crown glasses that lie at the extreme edge of the Ndd glass chart, notably  the “FK” and “PK” glasses. These glasses can offer very favorable relative partial dispersions  while very desirably providing a large separation in apochromat element V-numbers. However, before adoption in the final design, their optical shop and field environmental characteristics need to be considered and appropriately managed.

In particular, some are “soft” making grinding and polishing to specications more challenging than normal crown glasses. Also, a higher than normal thermal expansion can lead to breakage in the optical shop, coating lab as well as in the instrument in the field.

Many of the glasses considered for the flint element of an apochromat, while exhibiting favorable relative partial dispersions, also have issues pertaining to their design, manufacture and use in the field.

The rare earth glasses, i.e., the lanthanum “LA” types,  can exhibit favorable relative  partial dispersions, but tend to be very hard and more difficult to polish to a smooth scratch-free condition. Some, which are located at the very high index end of the glass chart, tend tend to be yellowish in transmission and show tendency toward staining.

The short flints, the “KZ” types, exhibit the very necessary “non-normal” relative partial dispersions for apochromat design, but these also show varying degrees of resistance to staining, some quite bad in this respect. Resistance to attack from moisture and acids encountered in the air, shop processing and hand contact can be insufficient in achieving and maintaining a smooth, clear optical surface, before and after application of any coatings.

Special glasses tend to be much more costly than ordinary optical glasses and not always  available in large sizes, available on a timely basis or with the index of refraction precision needed.

In periscopes with many apochromat relay lenses, availability of lens blanks with the required refractive  index homogeneity and constancy in value from one lens blank to another, can dictate the need for melt data redesign. This is a very large issue in terms of continuing design cost, glass ordering, scheduling,  handling procedures and on-going  record keeping. Vigilance to avoid mix-ups in  the optical shop through factory stores and field inventory must be continually maintained, as well as configuration management records of the periscopes in the field. The rationale for melt data redesign of an apochromatic  periscope  is discussed in the paper, “Importance of Melt Data Redesign in Submarine Periscopes”.

Apochromatic Periscope Summary

Traditional periscope design employs achromatic lenses of ordinary crown and flint glasses.  While time tested to be satisfactory for eyepiece viewing,  traditional periscopes suffer from a large amount of secondary spectrum, a  residual chromatic aberration.  This aberration seriously impacts image quality over the broader spectrums required by photographic films and electronic sensors.

Apochromatic periscope design employs special glasses with non-normal dispersion which permit the correction of secondary spectrum to a large extent. Beyond the application of special glasses, the secondary spectrum can be further reduced by reconfiguration of the layout of the objective lens and relay lenses. This also allows the further correction of spherical aberration due to the larger F-numbers which can be employed.  The resultant effect is that the apochromatic design yields nearly diffraction limited imagery from 450-700 nm wavelength. Caution in the use of the special glasses should be exercised since, as noted above, problems can occur  in glass procurement,  lens manufacture, logistics,  and instrument use.