Collor-Corrected Mangin Mirror Seymour Rosin and Max Amon Extension of the principle of the Mangin mirror to a color-corrected version yields designs superior to the equivalent paraboloid. This is true theoretically with regard to coma correction and practically with regard to ease of fabrication, particularly for high relative apertures. Examples are given of designs with low secondary spectrum and one with zero third-order coma. One of the most important tasks of classical optics is to form a sharp image of a small distant object such as a star. Since the time of Descartes, it has been known that a single paraboloidal reflecting surface has the proper geometrical shape to accomplish this purpose and affords a simple solution to the problem. However, the paraboloid has certain deficiencies. Theoretically, its field of view is coma-limited to small values. Practically, since the surface is nonspherical in shape, it is difficult to produce, and this difficulty in- creases severely with the relative aperture; thus, for steep paraboloids (low f numbers), the skill and expense required often become prohibitive. Almost a century ago, the French colonel, Mangin, proposed an imaging system composed of spherical surfaces in the form of a concave meniscus silvered on the back, as shown in Fig. 1. If one chooses the radius of curvature of the entering (and also existing) surface to be approximately two thirds that of the reflecting surface, the undercorrected spherical aberration of the latter is neutralized by the two passes through the overcorrecting front surface (principally by the first) and the resultant point image can be made quite good for any chosen color. The term "Mangin Mirror" is well known and is indeed part of the folklore of optics, its meaning being stretched to cover any rear surfaced spherical reflector. However, if we examine modern texts we find reference to this system either nonexistent or perfunctory. 1-4 The only serious application for the Mangin cited seems to be for such noncritical items as searchlights or projection condensers. In all fair- ness, however, Jacobs 2 does state that it is an "interest- ing optical system that may receive further develop- ment". A third-order analysis of the classical Mangin reveals not only correction for spherical aberration, but a The authors are with the Kollsman Instrument Corporation, 80-08 45th Avenue, Elmhurst, New York 11373. Received :19 October 1966. considerably reduced value (about 40%) for the coma coefficient compared with that of a paraboloid of equal aperture and focal length. Examination of Fig. 1 reveals that the only apparent degrees of freedom in the design are (1) the thickness and (2) the index of refrac- tion. We have analyzed these variations and have found that while the required radius of curvature of the front surface to achieve spherical correction may change, the 400 value for the coma coefficient as compared with that of the equivalent paraboloid is not much affected. There seems to be no basis for Strong's statement 4 that the coma may be corrected as well as the spherical aberration. However, the classical Mangin is not corrected for primary color aberration. The light rays traverse two negative singlet lenses, one before and one after reflec- tion, and these cannot help but impress color overcor- rection at the image plane. The performance of an f/1.75 classical system of 5.0 units aperture as ray traced is given in Fig. 2. Here the abscissas represent displacement of each chosen ray from the center of the aperture and ordinate its displacement in the image plane from true focus. We were interested in per- formance over a spectral range extending from ap- proximately 550 mu4 to 1100 mu4, corresponding to the sensitivity of a particular solid state sensor. The wave- lengths are as indicated, with XI representing a value near the center of the range. Examining the graph at 00, we see the excellent spherical correction for Xi, but X 2 and X,3 are seriously defocused. Turning our attention to the 1° off-axis trace, we see in the merid- ional trace the symmetrical, concave upwards, curve for Xi characteristic of coma and its value which is somewhat less than half that of the equivalent parab- oloid. However, when X 2 and X 3 are also considered, the coma is completely overshadowed by the color aberration. It is clear that if the Mangin is to be useful for critical application over any reasonable wavelength interval, it must be color corrected. To this purpose, the Man- gin may be thickened to the point where an interior May 1967 / Vol. 6, No. 5 / APPLIED OPTICS 963
Transcript
Collor-Corrected Mangin Mirror
Seymour Rosin and Max Amon
Extension of the principle of the Mangin mirror to a color-corrected version yields designs superior to theequivalent paraboloid. This is true theoretically with regard to coma correction and practically withregard to ease of fabrication, particularly for high relative apertures. Examples are given of designswith low secondary spectrum and one with zero third-order coma.
One of the most important tasks of classical opticsis to form a sharp image of a small distant object suchas a star. Since the time of Descartes, it has beenknown that a single paraboloidal reflecting surfacehas the proper geometrical shape to accomplish thispurpose and affords a simple solution to the problem.However, the paraboloid has certain deficiencies.Theoretically, its field of view is coma-limited to smallvalues. Practically, since the surface is nonspherical inshape, it is difficult to produce, and this difficulty in-creases severely with the relative aperture; thus, forsteep paraboloids (low f numbers), the skill and expenserequired often become prohibitive.
Almost a century ago, the French colonel, Mangin,proposed an imaging system composed of sphericalsurfaces in the form of a concave meniscus silvered onthe back, as shown in Fig. 1. If one chooses the radiusof curvature of the entering (and also existing) surfaceto be approximately two thirds that of the reflectingsurface, the undercorrected spherical aberration of thelatter is neutralized by the two passes through theovercorrecting front surface (principally by the first)and the resultant point image can be made quite goodfor any chosen color. The term "Mangin Mirror" iswell known and is indeed part of the folklore of optics,its meaning being stretched to cover any rear surfacedspherical reflector. However, if we examine moderntexts we find reference to this system either nonexistentor perfunctory. 1-4 The only serious application forthe Mangin cited seems to be for such noncritical itemsas searchlights or projection condensers. In all fair-ness, however, Jacobs2 does state that it is an "interest-ing optical system that may receive further develop-ment".
A third-order analysis of the classical Mangin revealsnot only correction for spherical aberration, but a
The authors are with the Kollsman Instrument Corporation,80-08 45th Avenue, Elmhurst, New York 11373.
Received :19 October 1966.
considerably reduced value (about 40%) for the comacoefficient compared with that of a paraboloid of equalaperture and focal length. Examination of Fig. 1reveals that the only apparent degrees of freedom in thedesign are (1) the thickness and (2) the index of refrac-tion. We have analyzed these variations and havefound that while the required radius of curvature of thefront surface to achieve spherical correction maychange, the 400 value for the coma coefficient ascompared with that of the equivalent paraboloid is notmuch affected. There seems to be no basis for Strong'sstatement4 that the coma may be corrected as well asthe spherical aberration.
However, the classical Mangin is not corrected forprimary color aberration. The light rays traverse twonegative singlet lenses, one before and one after reflec-tion, and these cannot help but impress color overcor-rection at the image plane. The performance of anf/1.75 classical system of 5.0 units aperture as raytraced is given in Fig. 2. Here the abscissas representdisplacement of each chosen ray from the center of theaperture and ordinate its displacement in the imageplane from true focus. We were interested in per-formance over a spectral range extending from ap-proximately 550 mu4 to 1100 mu4, corresponding to thesensitivity of a particular solid state sensor. The wave-lengths are as indicated, with XI representing a valuenear the center of the range. Examining the graphat 00, we see the excellent spherical correction forXi, but X2 and X,3 are seriously defocused. Turning ourattention to the 1° off-axis trace, we see in the merid-ional trace the symmetrical, concave upwards, curvefor Xi characteristic of coma and its value which issomewhat less than half that of the equivalent parab-oloid. However, when X2 and X3 are also considered,the coma is completely overshadowed by the coloraberration.
It is clear that if the Mangin is to be useful for criticalapplication over any reasonable wavelength interval,it must be color corrected. To this purpose, the Man-gin may be thickened to the point where an interior
Fig. 2. Graph of ray trace results, classical Mangin.
964 APPLIED OPTICS / Vol. 6, No. 5 / May 1967
RadiusSurf. No.
(1)(2)
(3)
-8.99
-13. 45
-8.99
- em.
r} } w
.000
FINO. = 1. 76
X
. , Xl--.- --I X.
3
KOLLSMAN MANGIN F/No. = 1.76
EFL = 8.780
BFL = 7.935
Surf. No. Radius
(1) -9. 435(2) 37. 500
(3) -14.357
(4) 37. 500
(5) -9. 435
Thickness
0.346
1.092
-1.092
-0.346
Nd
1.6134
1.6134
-1. 6134
-1. 6134
Fig. 3. Layout and design of color corrected Mangin.
KOLLSMAN MANGINAPERTURE=5.0 f/1.75
IX~x3X3
/I/2
. .. . - I ./
KOLLSMAN MANGIN- - - - PARABOLOID
CODE WAVELENGTH (MICRONS)
XI .7682X 2 .5876X3 1.0140
'1 XX2X3'2
l.o
\ ~~~~~~~~~/\ \ /
. I - I i
-2.5 -1.5 -0.5
X3
0.5 1.5 2.5
07 S
I ~~~~<.11 XX2
0 10 2.0
HEIGHT ON ENTRANCE PUPIL
0.0e
HEIGHT ON ENTRANCE PUPIL
Fig. 4. Graph of ray trace results of system of Fig. 3.
May 1967 / Vol. 6, No. 5 / APPLIED OPTICS 965
V Glass
57.28
43.91
43.91
57.28
SK 19
KZFS4
KZFS4
SK 19
.004 -
.003 -
.002 -
.001 -
.004 -
.003-
.002-
.001-
.000
.001-
I
I
I
.000
-.001 1
KOLLSMAN MANGINAPERTURE=2.5 f/1.25
KOLLSMAN MANGIN---- PARABOLOID
CODE WAVELENGTH(MICRONS)|
Xl .7682X2 .5876X3 1.0140
0.5'
11 "I1 ' X )
XI~~~~~~~1X-I. 11 -'- X2-
-.75 -.25
HEIGHT ON ENTRANCE I
X 2.25 .75 1.25
'UPIL
Fig. 5. Graph of ray trace results of constructed system.
AIR SPACED KOLLSMAN MAINGEx F/No. = 3. 2
EFL = 16. 019
BFL = 15.301
Surf. No.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Radius
-12. 009
18.979
19. 198
-20. 700
19. 198
18. 979
-12. 009
Thickness
0.340
0. 100
0.700
-0.700
-0. 100
-0.340
Nd
1. 6134
1.0
1.6134
-1. 6134
-1.0
-1. 6134
-1.0
Vd
57.28
43.91
43. 91
57.28
Fig. 6. Layout and design of coma corrected Mangin.
surface may be included, thus dividing it into positiveand negative components as shown in Fig. 3. Sincethe over-all lens form is a negative meniscus, the positivecomponent must be made of glass of higher dispersion(lower Abbe number) than the negative.
As a first try, we chose glasses of matching index ofrefraction and different dispersions. In addition, weendeavored to make the choice involve glasses whosepartial dispersions were not too far apart. Glasseswhich met this general criterion were SK19 for the
966 APPLIED OPTICS / Vol. 6, No. 5 / May 1967
.002-
.001 -
.002-
.001-
0.35
-1.25
.25 .75 1.25
HEIGHT ON ENTRANCE PUPIL
0.0-
Glass
SK 19
KZFS4
KZFS4
SK 19
* -- t 1 1 1
.QUO I l I
000; 1 1 1 1 1
)II XIX3
KOLLSMAN MANGIN ( AIR SPACED)APERTURE=5.0 f/3.2
KOLLSMAN MANGIN- -- - PARABOLOID
//
//
/-I
NNN
I// CODE WAVELENGTH (MICRONS)
Xl .7682X2 .5876X3 1.01 40
1.0e I X
X2
.003- \\
.002 /
.001 N -I
N~~~~~~ /
-.001 _~X
-.002 .7
.001
.000 i i I I 5 @ X3
1.0
0 10 205.
0.01
-.001
-2.5 -1.5 -0.5
I
X2 000.5 1.5 2.5
Fig. 7. Graph of ay trace results of system of Fig. 6.
negative component and KZSF4 (Schott glasses) for thepositive. Since the indices are alike, we could obtaincolor correction through simple alteration of the interiorsurface, without at the same time affecting sphericalcorrection. The resultant design is also shown in Fig.3.
Figure 4 gives the graph of the ray trace results on thisconfiguration, which may be compared directly withthe uncorrected Mangin of Figs. 1 and 2, since theaperture and f/number are identical in the two cases.The improvement is quite dramatic. Axial perfor-mance is excellent, and off-axis performance is more thantwice as good as the equivalent paraboloid.
Figure 5 gives the results for a similar Mangin of6.35-cm aperture working at f.25. The graph isnormalized to an aperture of 2.50 units; therefore, thefocal plane displacements read directly in inches, al-though no actual reference is made to the units in-volved. These results are included because the systemwas actually constructed for a star tracker application.
We turn now to a discussion of the coma. It will berecalled that the coma coefficient of the uncorrectedMangin was about 40% of that of the equivalent pa-raboloid. This value is insensitive to changes in thick-ness. Inasmuch as the two components of our color-corrected Mangin have the same index of refraction,the 40% figure is retained, and the graphs of Figs. 4and 5 bear this out. However, the introduction of theinterior surface imparts an additional degree of freedomto the system, which may be used for the elimination ofcoma entirely. First, if the indices of refraction of thetwo components are allowed to be different, coma andspherical contributions arise from the interior surface.
Preliminary third-order calculations indicate that comacan be corrected or even reversed in sign if the negativeelement has a substantially lower index than the posi-tive. We did not carry these calculations to conclu-sion since no immediate need existed. Further studywill be made making use of Spencer's ACCOS auto-matic lens design program.
Another degree of freedom for the correction of comawhich we found more immediately easy to apply wasthe introduction of an air space between the com-ponents. We reverted to the original glasses. Themethod of design is identical to that used in designingan air spaced refracting doublet. First, the separatepowers of the two components were adjusted to correctprimary color. This correction is retained throughoutsimple bending. Then the bending of the two com-ponents affords two degrees of freedom that are utilizedto correct spherical aberration and coma. Contraryto the situation for an air spaced doublet, however,this bending produces large changes in the over-allfocal length, since the bending affects the radius ofcurvature of the reflecting surface.
Application of this process resulted in the config-uration of Fig. 6, where the third-order coefficients forspherical aberration, coma, and color are all zero.In this graph the aperture was held at 5.0 units, but thefocal length changed because of the above bendingprocess so that the system now worked at f/3.2. Raytrace results are shown in Fig. 7 and indicate a dis-appearance of the coma. Residual higher-order aberra-tions are present which no doubt can be reduced byrefinement of the design. We plan to apply ACCOSin this case also.
May 1967 / Vol. 6, No. 5 / APPLIED OPTICS 967
.002
.001-
References]. Design of Fire Control Optics, Ordnance Corps Manual
sanne Gen. Sec., Eidgenossische Landestopographie,Sefligenslr. 264, Bern-Wabern, Switzerland
10-14 SPSE Ann. Conf., Sheraton-Boston, Boston, Mass.SPSE, Main P.O. Box 1609, Washington, D.C.
23-28 ASTM 71st Ann. Mtg. and 18th Materials TestingExhibit, Statler-Hilton Hotel, San Francisco, Calif.1916 Race St., Philadelphia, Pa. 19103
September1-7 4th European Regional Conf. on Electron Micro-
scopy, Rome D. S. Bocciarelli, Istituto Superioredi Sanita, Viale Regina Elena 299, Roma, Italy
3-7 23rd Ann. Symp. on Molecular Structure and Spec-troscopy, Columbus K. N. Rao, Dept. Phys., OhioState U., 174 W. 18th Ave., Columbus, Ohio 43210
October8-11 Optical Society of America, 53rd Ann. Mtg., Pitts-
burgh Hilton M. E. Warga, OSA, 1155 16th St.,N.W., Washington, D.C. 20036
20-25 104th Semiann. Conf. SMPTE, Washington, D.C.Ex. Sec., 9 E. 41st St., New York, N.Y. 10017
28-31 ISA Ann. Instrument-Automation Conf. and Exhibit,Coliseum, New York, N.Y. D. R. Stearn, ISA,Penn Sheraton Hotel, Pittsburgh 19, Pa.
Autumn 7th Internatl. Conf. on Microwave and OpticalGeneration and Amplification, Hamburg Nach-richtentechnische Gesellschaft im Verband DeutscherElektrotechniker
Optical Society of America Spring Mtg., Washington,D.C. M. E. Warga, OSA, 1155 16th St. N.W.Washington, D.C. 20036
SPSE Ann. Conf., New York City SPSE, Main P.O.Box 1609, Washington, D.C.
107th Semiann. Conf. SMPTE, Chicago D. A.Courtney, 9 E. 41st St., New York, N. Y. 10017
October? Optical Society of America, 55th Ann. Mtg. M.
E. Warga, OSA, 1155 16th St. N.W., Washington,D.C. 20036
4-9 108th Semiannual Conf. SMPTE, New York D. A.Courtney, 9 E. 41st St., New York, N.Y. 10017
21-24 SPSE Ann. Symp., Wash., D.C. SPSE, Main P.O.Box 1609, Washington, D.C.
Summer Clinicin
X-ray Spectroscopy
The annual clinic in x-ray spectroscopy will be con-ducted from 5 June to 10 June 1967 at the x-raylaboratories of the State University of New York atAlbany. Registration is open to anyone engagedin the field of x-ray spectroscopy. No previousknowledge or experience will be assumed. Morningsessions will be devoted to lectures and discus-sions, emphasizing fundamental principles andtheory; afternoon sessions will be devoted tolaboratory experiments, illustrating the lectures.Basic methods including sample preparation, ac-cumulation of standards, interpretation of dataand current instrumenttechniqueswill be stressed.The cost of the clinic is $250.00 payable in ad-vance. Direct inquiries to Henry Chessin, PhysicsBuilding, Room 214, State University of New Yorkat Albany, 1223 Western Avenue, Albany, NewYork 12203.
