Testing of flat optical surfaces by the quantitativeFoucault method

Maria C. Simon and Juan M. Simon

The complete theory of measurement of optical flat mirrors of circular or elliptical shape using the quantita-tive Foucault method is described here. It has been used in C6rdoba since 1939 in a partially intuitive butcorrect form. The surface, not yet flat and, at times, astigmatic, is assimilated to the sum of a spherical plusa cylindrical dome. The errors of the three possible ways of reckoning are calculated.

1. Introduction

Plane surfaces play an important role in optics. Theproduction of medium and large surfaces by means ofgrinding and figuring is one of the most delicate anddifficult tasks. When shaping a concave or convexsurface a tolerance of millimeters and even centimetersin the radius of curvature is allowed. The opticianchooses a suitable radius of curvature between limits.In the case of a plane mirror there is no choice. Theradius of curvature must not be less than 500 km to havean acceptable plane mirror. That is, the distance be-tween astigmatic images [Eq. (16)] has a tolerance of afew hundredths of millimeters, which approaches thelimit of accuracy of measurement. The quantitativecontrol of its shape is important for finished surfaces,but it is even more important for a surface which isbeing figured when it is almost but not yet plane.

A finished plane surface can be verified by means ofa master flat, if you have one, using interferences.However, to verify the master flat itself, either two othermaster flats of equal size or larger, or another testingmethod, are required.

There is another method: the quantitative Foucaultone: it uses a master spherical mirror of a little largerdiameter (Fig. 1). A master spherical mirror is rela-tively easy to build with an accuracy of a few hundredthsof a wavelength.

In the optical shop before reaching the final planeconcave and convex surfaces, circular zones and oftenastigmatic or distorted surfaces are necessarily ob-tained. The last ones are difficult to avoid if the plane

The authors are with Universidad de Buenos Aires, Facultad deCiencias Exactas y Naturales, Departamento de Fsica, Ciudad Uni-versitaria, Pabellon 1, Nnez, Buenos Aires, Argentina.

Received 17 January 1977.0003-6935/i8/6161-0i32$0.50/0.© 1978 Optical Society of America.

is of elliptical contour, such as the flat diagonals ofNewtonian telescopes. The quantitative methodshould be able to measure and calculate the sphericalaberrations and the astigmatism zone by zone. If thisprocedure is not followed the risk of retouching thewrong place and of ruining the surface is great.

The use of the Foucault test to verify flat mirrorsusing a master spherical was done by Ritchey,1 but onlythe introduction of the zonal Foucault test by Gaviola2and Couder3 allowed the determination of the profileof the mirror.

The quantitative Foucault method was used by E.Gaviola and R. P. Platzeck in the C6rdoba Observatorybetween 1938 and 1942 to figure the Newtonian planemirrors of the Large Reflector of Bosque Alegre. Arecent publication,4 which considered the problem ina partial way, induced us to publish the completetheory.

11. Calculation of the Surface's Form in Case It hasSymmetry of Revolution

The measurement of a flat mirror can be carried outwith the quantitative Foucault method by observing thevirtual image of a master spherical mirror reflected onthe plane to be measured. The elements are disposedas shown in Fig. 1. Both mirrors have circular contours;the semidiameter of the plane one is r, and that of thespherical mirror is rR/D or larger.

The master spherical is supposed to be perfect. If itis not, its form can be measured and taken into ac-count.

The mirror to be measured has a 450 inclination inrespect to the axis of the spherical mirror and to the axisof the direction of observation. On account of such aslope, the plane mirror cuts an elliptical contour; andthe virtual image in the master spherical has a majorsemiaxis (see Fig. 1)

a = r(R/D) (1)

132 APPLIED OPTICS / Vol. 17, No. 1 / 1 January 1978

The variations of the sagitta of the image mirror dueto the sagitta of the mirror to be measured are given bythe difference of the optical paths between a perfect

U T y Y plane and the mirror of sagitta h. From Fig. 2, one

I obtains

bfb = hV2; b/a = ha/2. (7)

Putting these quantities into Eqs. (5) and (6), thevariations of the radius of curvature are

Fig. 1. Mirror positions and coordinates system.

1 1

4V/2D 2

bS = r2 h;

2V/2D2____h.

r2

From these relations two expressions for h follow:

h = r24V/2D 2

r2h = 6T.

2,\/2D 2

Thus, if 6S and 6T are measured, the sagitta of themirror can be determined. From Fig. 1 one obtains

Fig. 2. Change of the virtual spherical sagitta because of the actual

shape of the plane. Path difference 2h/V< = VO7h = bf.

and a minor semiaxis=R rR

b = r-sin45' D \/2D

where S and T are the positions of the astigmaticalimages, and U is the position of the centers of curvatureof the virtual spherical mirror, if the plane is perfect.

As the position of U cannot be fixed until the planeis perfect, h is determined as the difference:

(2)

where R is the radius of curvature of the sphericalmirror and

D = R -E,

E being the distance between the centers of both mir-rors.

If the plane mirror is perfect, the radius of curvatureof the virtual spherical is the same as that of thespherical mirror. The sagitta of the virtual sphericalwill be different for both axes, and their values are

fb = b2 /2R = r ;

fa = a2/2R =r2R-

(3)

If the mirror to be measured is not plane, but sphericaland with sagitta h, the radius of curvature of the virtualmirror will suffer a variation 6S in reference to the minor

axis, and 5T in reference to the major axis. The corre-sponding variation in the sags are obtained by differ-entiation of Eqs. (3) and (4) in respect to R.

Denoting iR = 5S for the shortened axis and eR = 6Tfor the other axis, we obtain

bfb = -2 6S; (5)4D2

3/a = 2 T. (6)4D 2

and A is obtained from the positions S and T because

A = S -7T.

If h is positive, the pseudoplane mirror is convex; ifit is negative, the pseudoplane mirror is concave. Fig-uring must continue until h is negligible (a few hun-dredths of a wavelength).

I11. Circular Zones of Aberration

If the mirror to be measured is not spherical but thesurface has symmetry of revolution, it is approximatedby spherical zones of different radii of curvature. Todo so, the plane mirror is divided in concentrical ringsof width brj by a double zone screen.

The screen may be put on the plane or on the spher-ical. In the first case measurements must be made witha Foucault apparatus without parallax, and the screenmust be glued to the plane. If the screen is set on thespherical mirror a certain parallax is admissible, but thezonal rings must have the elliptical form of the virtualimage of the plane on the spherical. Ellipses must havedifferent separations on the right and on the left becauseone side of the plane is nearer the spherical mirror thanthe other and because of the variation in slope.

1 January 1978 / Vol. 17, No. 1 / APPLIED OPTICS 133

D- R-E

2

0 R

(8)

(9)

(10)

(11)

bS=S- U,

6T= T- U,

(12)

(13)

A = S - T.

Subtracting Eq. (9) from Eq. (8) and detaching h,

h= A;2v2D 2

(15)

(16)

1

(14)

Differentiating Eqs. (10), (11), and (15) in respect tor, the following expressions for the increment of heightover each zone bhj are obtained:

bhj = _L2a_ bSj;V 2D2 2 ;

V\ 2D 2

(3j = rri iv\'2D2

1

(17)

0s = 2 T. (26)

Using Gauss's law of propagation of errors results in:(1) the error of 1/2 Sj:

a = (3 n S2 + T2) 1; (27)

(18) (2) the error of Tj :

(19)

As was mentioned above, only Eq. (19) could be usedto determine hj because bSj and Tj cannot be deter-mined, taking into account that the position of U isunknown. But that position may be calculated for eachzone. Thus, from Eqs. (8) and (9), this follows for the. zone:

Sj = 2Tj,

which with Eqs. (12) and (13) gives

UJ = 2Tj,-Sp.

(20)

(28)(3terof ( 2 + 12)1 2

(3) the error of Aj:" = (CS2 + T2)1/2.

(29)

With Cs = 2 T, the previous expressions are reducedto

/n+41/2a = n4) UT; (30)

(31)

(21)

These values differ between them on account ofmeasurement errors; for, if U is the position of thecenter of curvature when the plane is perfect, U shouldbe the same for every zone. Therefore, Eq. (21) is usefulas internal control of measurements. Besides from allthe values of U a mean value can be obtained:

- nU = - FZ Uj,

nj=i(22)

where n is the number of zones. With this value of U,we get

bSi = Si - U, (23)6TJ TJT-U; (24)

and from Eq. (16) it follows that

A = S - T . (25)

Thus three expressions for bhj are available.Errors in the three cases will be analyzed to decide

which of the three expressions is better to use.

IV. Analysis of Errors when Measuring a Surfacewith Revolution Symmetry

To know the difference between the errors of hjcalculated with Eqs. (17), (18), or (19), it is enough tocalculate the errors of (1/2)6Sj, 65Tj, and A, for the factorrj brj/V2D 2 is the same one in the three cases, and itcontains only screen errors. The magnitudes measuredare S and T, and the corresponding errors will becalled as and T.

As the errors as and T are inversely proportional tothe width of the zone and to the distance from the zoneto the center of the mirror, the zones of the screenshould reduce their width proportional to their distanceto the center of the mirror to obtain the same error forevery zone and to make the increments hj comparable.On account of the mirror's 45° slope, the width and thedistance of the zones of the shortened axis to the centerof the mirror are reduced by a factor /2 in respect tothe zones of the major axis. Therefore,

a-" = /5UT. (32)

Comparing the expressions of a, A', and or", we observethat for n > 1, a and a' are smaller than a", as we ex-pected, because the error of U is less than the error ofone measurement. If n = 1, no mean value can be cal-culated; and all the cases are reduced to the last one. Itis thus convenient to calculate the mirror profile usingeither Eq. (17) or Eq. (18).

V. Astigmatism and Sphericity

Plane mirrors often present astigmatic errors addedto spherical aberration. Astigmatic errors are practi-cally unavoidable during the optical figuring of planemirrors of elliptical contour (Newtonian diagonals).How are astigmatic errors detected? When we noticethat shadows on the mirror move transversally, or inpart transversally, when the knife-edge is advanced wemust suppose the presence of astigmatism. To confirmor deny it Aj [Eq. (25)] is measured in several azimuthalpositions of the plane mirror turning it around its axis.There must be at least three positions; four or more arebetter. If the values of Aj remain constant withinmeasurement errors, the mirror has no astigmatism. IfAj varies with the azimuth, astigmatism is present. Inthis case azimuths of major and minor variations areregistered as positions of the astigmatic axis. Thescreen of zones is placed according to these axes.

We measure in two normal azimuthal positions of theflat mirror and, each time, along the two axes (verticaland horizontal) defined by the screen of zones. Thusthe necessary data to calculate sphericity and astig-matisms are obtained.

Measurements and calculations are now more com-plicated. However a good approximation can be ob-tained if it is supposed that each ring-shaped zone ap-proaches the shape that results from the superpositionof a spherical and a cylindrical dome.

The radius of curvature variations will be, then, theaddition of the variations due to the spherical defor-mation and the cylindrical one.

134 APPLIED OPTICS / Vol. 17, No. 1 / 1 January 1978

C = (n 4 12 CT;

n -)

The variation according to the minor and major axesdue to the spherical deformation, a function of the in-crement of sagitta, is obtained by differentiating Eqs.(10) and (11) with respect to r.

Introducing the subindex e to point out that it onlyrefers to the spherical deformation and calling

3= , (33)V\2D2

one obtains

(Sej = (2/g) (3hej; (34)

5 Tej = (11g) (hej. (35)

The radius of curvature variation due to the cylindricalsurface will depend on the orientation of the cylinder'saxis.

Let us see the two cases corresponding to the twopositions in which the mirror is measured.(1) The cylinder's axis is parallel to the minor axis of thevirtual spherical mirror:

(36)(silci = 0;

6T IIci = (1/g) hcj.

Subindex c indicates that only the cylindrical defor-mation is meant.(2) The cylinder's axis is perpendicular to the minor axisof the virtual spherical mirror:

(S±Ij= (2/g) (hcj; (38)

6T 2j = 0. (39)

The total variation of the radius of curvatur is obtainedby adding Eqs. (34) and (35) to Eqs. (36) and (37) andto Eqs. (38) and (39), respectively:

BSIl = (2/g) (3hej; (40)

bTl= (1/g) hej + (1/g) hcj; (41)

BS j = (2/g) hej + (2/g) hcj; (42)

6T 7 = (1/g) hej. (43)

Subtracting Eq. (41) from Eq. (40) and Eq. (43) fromEq. (42), the differences between radius of curvaturevariations are obtained:

Alj = (1/g)(hej - (hj); (44)

A Ij = (1/g)(6hej + 26hj). (45)

From these equations, bhcj and 3he can be detached asfunctions of the directly measurable parameters A 11j and

(hcj = (g/3)(A 1 j -Aj);

(hej = (3) (A1 j + 2iIj).

(46)

(47)

The total increment of height for each zone is ob-tained by adding the two previous relations.

being SIj and T 1j, the saggital and tangential imagepositions when the cylinder's axis is perpendicular tothe minor axis of the virtual spherical mirror, and S1jand T11j are the positions of those images when the cyl-inder's axis is parallel to the shortened axis.

Replacing Eqs. (50) and (51) in the expressions of bhjand bhj, one obtains

(hej = (3)[(SIj - TIj) + 2(SIj -Tllj)];

bhj = (g/3)[2(SLj - Tj) + (Sl1 - Tllj)]-

(52)

(53)

Another way of obtaining hej results from Eqs. (40) and(43):

(54)

(55)

(hej = (g/2)bSiij;

(hej = gb7Tj.

From Eqs. (41) and (42) bhj is obtained:

(hj = g6TIj ;

6hj = (g/2)6SIj.

To determine the radius of curvature variations wemust first obtain the position of the center of curvaturecorresponding to the virtual spherical mirror, if themirror to be measured is perfectly flat. When themirror is rotated its position changes, so the positionsS, T, and U vary; there are then two different values forthe center of curvature position of the virtual sphericalmirror: U11and UI. Therefore, the radius of curvaturevariations are

5Siij = Sluj - U11;

bTIIj = T llj - Ull;

BSIj = SIj - U1;

6T i j = Tj- U1.

(58)

(59)

(60)

(61)

The relations that give U11and U1 in function of theimage positions are obtained as follows: for U11 Eqs.(46) and (47) are inserted in Eqs. (40) and (41):

(SlIj = 2/3(AIj + 2AIIj);

6TIIj = 1/3(2A j + A11j).

(62)

(63)

Replacing all the parameters by their respective ex-pressions given in Eqs. (50), (51), (58), and (59), bothrelations show the same expression for U11,

Ull = 4/3TIIj - 1/3SIlj + 2/3(T j -S ). (64)

In the same way UL is obtained. Putting Eqs. (46) and(47) in Eq. (42) we obtain

BSIj = 2/3(Alj + 2A1 j);

6T j = 1/3(AI 1 j + 2A11j);

that with Eqs. (50), (51), (60), and (61) gives

(65)

(66)

(56)

(57)

(hj = (3hej + hcj;

bhj = g/3)(2AI1 j + A 11 ),

where

Alj = Sjj - Tj,

(48) UIj = 4/3T j - 1/3S 1j + 2/3(Tllj - SIij). (67)

(49) From Eqs. (64) and (67) we can see that a value U111

and another UIj are obtained for each zone. Thesevalues differ for different zones only due to experi-

(50) mental errors. Thus such relations are useful as an

1 January 1978 / Vol. 17, No. 1 / APPLIED OPTICS 135

AIlj = Sllj -Tlij, (51)

internal control of measurements, and the averages tocalculate the radius of curvature variations can be cal-culated as

- inU1 l = - E U11i,n i=1

U1=E uli.n i=1

U = (5/ 3 ) UT-

_ (2) The increments of height are obtained withU11:

(68)

(69)

Thus, using radius of curvature variations given inEqs. (58)-(61), from Eqs. (54)-(57) the increments hejand hj are given by

(hej = g/2)(S11j - Uii); (70)

(hej = g(Tlj - U); (71)

ahj = g(T11j - U 11); (72)

(3hj = (gI2)(SIj - U ). (73)

Equations (52), (70), and (71) provide three differentways to calculate bhej, and Eqs. (53), (72), and (73) givebhj. In the following section the analysis of the errorsfor the three cases is made.

VI. Analysis of the Errors When Measuring anAstigmatic Surface

The magnitudes directly measured are the positionsof the astigmatic images for two orientations of themirror to be measured, which is perpendicular betweenthem.

The procedure of rotating the mirror introduces achange in its position by which only the pairs (Sij, Tj)and (Ij, Tj) are well determined.

As the relations deduced in the previous paragraphtake this fact into account, the errors as and T, cor-responding to the positions Sj, S 1j and Tj, T 1j, re-spectively, include no error due to the mirror rotationprocedure.

To calculate how the errors as and 0-T influence theerrors of hej and hj, Eqs. (52), (53), (70), (71), (72), and(73) are separated into two factors: the factor g = [(rj* 6rj)/(V'2D2 )] is the same in every case, only includesthe errors of the screen, and will not be considered in thecalculation that follows. The remaining factors will becalled fej and fj. The increments hej and hj will thusbe expressed:

(hej = gfej; (74)

(hj = gfj. (75)

fej and fj will be given in each case comparing Eqs. (74)and (75) with the corresponding expressions of hej andbhj given in the previous sentence.

Applying Gauss's law about propagation of errors tothe factors fj and fj in each case the following is ob-tained:

(1) The increments of height are directly obtainedfrom the position of astigmatic images:

a, ((5/9) s 2 + (5/9) T2

)1

/2, (76)a ((5/9) U

2+ (5/9) T

2)1

/2. (77)

With a = 2 T they give

ae = (5/3)aT, (78)

[4 ( 9n) 9n ]T

C = [ (1--4) IT2 + - S2] 2

With as = 2 T they give

,,e = (1 + 16) 1/2r

(80)

(81)

(82)

(83)" = (+ 16)1/2 T.

_(3) TheU1 :

increments of height are obtained using

(84)

(85)

ae= [(1 ) T2+ aS2] 1/2

a' = [-1 +-)1 aS2 + 9 2] 1/2

With as = 2 T they give

ae = (1 + 6) 1/2T

a = ( 1 + 16)1/2 T9nT

(86)

(87)

Comparing the results obtained in the three cases weobserve that the errors are the same in the last two andthat they decrease when n increases. If n = 1, a anda are equal to o' and a'; but for n > 1, ae and a arelarger than ae' and a'. For that reason the mirror'sprofile should be calculated using Uii or UI.

VII. Measurement of a Plane Mirror with EllipticalContour

In general, astigmatic surfaces appear when workingwith pieces of noncircular contour, for example, planemirrors of elliptical contour of Newtonian telescopes.The relation of the axis of those mirrors is b = a/v'2.That is why they are seen with circular contour when themajor axis is inclined 450 with respect to the visual line.Thus, it is enough that the diameter of the sphericalmirror is equal to the minor axis of the plane multipliedby RID (Fig. 1).

Measurements are made dividing the flat mirror incircular zones, with the border ones remaining incom-plete. The cylindrical axis of the dome is chosen par-allel to the minor axis of the plane. Setting the mirrorso the minor axis is parallel to the shortened axis, thepositions T11j and Sj of the complete zones are mea-sured.

Then the mirror is turned 900, and Tj (belongingto the complete zones) and Sj (belonging to the com-plete and incomplete ones) are measured.

The increments of height hej and hj of the completezones can be calculated by means of any of the formulasmentioned in the previous paragraph, where the aver-ages U1 and U are only made upon the completezones. For incomplete zones only Sj can be measured,so bhj can only be determined by Eq. (73).

136 APPLIED OPTICS / Vol. 17, No. 1 / 1 January 1978

(79)

Table 1. Measurement and Calculation of the Shape of an Elliptical Contour Plane Mirror (33 cm X 46 cm)a

Zone Eq. Units 1 2 3 4 5 6 7

External cm 5.3 8.8 12.4 15.0 17.8 20.2 23.0

radiusS, ~~- 10 3 CM -123 -44 6 32 39 26 0

T,± - 10 3 CM -13 15 6 -52

S11 - 10 3 CM -28 95 125 28T

11 - 10 3 CM 57 89 131 134

U± (67) 10-3 CM - 31 10 -9 U±=10 X 10-3 CM

U11 ~~(64) 10-3 cm - 126 133 113 =1 124 X 10-3 CM

Si - U± - 10-3 CM -133 -54 -4 22 29 16 -10

721-U± ~~ ~- 10 3 CM -23 5 -4 -6

Sii - 11- 10-3 cm -152 -29 1 -96T1- U11 - 10-3 CM -67 -35 7 10

g/X (33) X/cm 0.658 1.174 1.764 1.71 2.04 2.36 2.68

M3 (73) X -0.044 -0.032 -0.003 0.019 0.030 0.019 -0.013

(3he (71) X -0.015 0.006 -0.007 -0.106

(3he (70) X -0.050 -0.017 0.001 -0.082

(3h (72) X -0.044 -0.041 0.012 0.017

Zb(h (73) X -0.044 -0.076 -0.079 -0.060 -0.030 -0.011 -0.024

163he (71) X -0.015 -0.009 -0.016 -0.122

163he (70) X -0.050 -0.067 -0.066 -0.148

16(h (72) X -0.044 -0.085 -0.073 -0.056

.. --.-. L ,f IT. --nr T,, fh- o.nfrnl 7nnp hacs not been used because the error in S and 72 is much larger in this zone thai

in the others. As g is smaller for the central zone, a larger error in S and T does not produce important errors in the mirror profile.

2 3 4 5 6 7

.Eq.72) (Eq.73)

1 2 3 4)l6^^^^^^^^^^^^^^^^^^^^^^^^^^^^I^I

\ ( Eq. 71)- 0.1 X

(Eq.70 1

Fig. 3. Profile of an elliptical contour plane mirror (33 cm X 46 cm)

of Newtonian focus of Bosque Alegre 154-cm telescope. The con-

tinuous lines correspond to the evaluation made in C6rdoba, and the

dotted lines correspond to the other possible evaluation.

In Table I and Fig. 3 the measurement of one of theplane mirrors of elliptical contour (33 cm X 46 cm),which was used as a Newtonial diagonal mirror for the154-cm telescope in Bosque Alegre, C6rdoba, is shown.The measurements are reelaborated here, according tothe complete theory. The continuous lines in Fig. 3show the results obtained by Gaviola and R. P. Platzeck[Eqs. (73) and (70)], and the dotted lines represent theother possible evaluation of the measurements [Eq. (71)and (72)]. An average of both curves is used during theshaping, but the separate drawing allows us to appre-ciate the self-consistency of the measurements.

V1II. Conclusions

Construction of plane mirrors using the quantitativeFoucault method can be facilitated with adequatemanagement of the data measured. Astigmatic mirrorsand those of elliptical contour can also be measured withthe accuracy required for telescopes and for masterflats.

We acknowledge E. Gaviola's suggestion about thesubject and the information about the experimentaldata that illustrate this paper.

References1. G. W. Ritchey, Smithsonian Contributions to Knowledge, No. 1459

(1904); reprinted in Sc. Am. Supp. (7 Jan. 1905); also quoted by

J. M. Pierce in Amateur Telescope Making (Scientific American,New York, 1962), Part 10, Chap. 3, p. 241.

2. E. Gaviola, J. Opt. Soc. Am. 26, 163 (1936).3. A. Couder, Rev. Opt. 16, 17 (1937).

4. C. J. Silvernail, Appl. Opt. 12,445 (1973).

1 January 1978 / Vol. 17, No. 1 / APPLIED OPTICS 137

76h

0

0.1 X.

A6h.

0

1

In

6