Optical applications of solid glass spheres

OPTICAL APPLICATIONS

OF SOLID GLASS SPHERES

PROEFSCHRIFT

TER VERKRUGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECH· NISCHE HOGES<.:HOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. R. KRONlG, HOOG-LERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE

SENAAT TE VERpEDIGEN OP WOENSDAG 18 NOVEMBER 1959,

DES NAMIDDAGS TE 4 UUR

DOOR

ADRIAAN WALTHER

NATUURKUNDIG INGENIEUR GEBOREN TE 'S·GRAVENHAGE

'959

GRAFISCH BEDRIJF AVANT! • DELFT

H/~

:<

!

.' I . • I /

. J

Dit proefschrift is goedgekeurd door de promotor Prof. Dr. A. C. S. van Heel

Aan ntijn ouders,

CONTENTS

Introduction; summary . . . 7 Part I. Geometrical optics of tbe spbere

1. Introduction . . . 9

2. Fundamental relations . . . . 10

3·. Paraxial optics of tbe spbere. 12 4. Tbird order aberrations of tbe sphere 17 a. Introduction . . . 17 b. Third order aberrations of concentric

systems . . . . . 20 c. Third order aberrations of tbe sphere 24 5. Tbe sphere in tbe class of thick single lenses 26 6. Van Leeuwenhoek's microscope . . . . . 35 Part Il. Physical optics of tbe sphere

1. Rigorous geometrical tbeory . . . . . 40 2. The diffraction pattern associated witb tbe

limiting ray. . . . 50

3. Theory of tbe rainbow . . . 57 Part III. Applications

1. The determination of refractive indices 60

a. Introduction . . . 60

b. Modified arrangement. 62

c. First experiments 64

d. Widtb and height of tbe slit 67

e. Furtber experiments . 72

f. Final remarks . . . . 75

2. Focusing and tbe measurement of large radii of curvature. . . . . 79

a. Focusing of collimators and telescopes 79 b. Tbe measurement of large radii of

cur-vature . . . 81 !l. Miscellaneous applications 85 Nederlandse samenvatting. References 88 90 91 Appendix

INTRODUCTION; SUMMARY

This thesis deals with the optical properties of solid glass spheres. Spherical surfaces are currently employed in opties; applications of the entire sphere however are very rare. The only wen known example is Van Leeuwenhoek's microscope.

In the first part of this thesis we discuss the properties of the sphere used as a lens. We have taken the position of the object and the stop to be arbitrary; also the refractive index is treated as a variabie. Moreover we have assumed that the light is subject to any number of internal reflections in between ente-ring and leaving the sphere. The derivation of the paraxial- and third order properties in this rather general case seems to be a tedious task. Fortunately we had some recent developments in the theory of third order aberrations at our Jisposal due to Korringa, Stephan and Brouwer. Their work greatly simplified our investigations.

The result of our calculations is rather disappointing: it ap-pears that the aberrations are always large. Yet we decided to publish these results; no so much because of their practical importance but rather as a demonstration how the new develop-ments in the third order theory can be put into practical use.

A few pages are devoted to Van Leeuwenhoek' s microscope. As a matter of fact Van Leeuwenhoek did not use spheres; his microscopes were provided with thick symmetrical single lenses slightly thinner than a sphere would be. Van Cittert raised the question why he did not use spheres; they are more easily made and Van Cittert suggests that the quality of the image might be better. This suggestion we have proved to be true; the difference appears to be inappreciably small however. Van Albada' s ass ertion of the sphere having the smallest spherical aberration of all thick single lenses we have proved to be wrong.

More interesting results are obtained when we approach the sphere from the viewpoint of physical opties. Here we arrive I\t the theory of the rainbow, that we have summarized in part two of this thesis. This theory dates back to Descartes and Newton; ft has been refined and completed by Airy and Stokes, about a hundred and twenty years ago. Mascart paid considerabie attention " to the rainbow; in his "Traité d'Optique", published in 1889, he treated it very extensivély. He mentioned a few laboratory ex-periments but he did not come to any practical" applications.

Applications are easily found. The rainbow can be imitated in the laboratory by putting a glass sphere in a parallel pencil

of light. Tbe direction in which the rainbow is observed depends on the refractive index of the sphere, a fact already proved by Newton. With this elementary consideration as a starting point we have developed a new way of measuring the refractive index of solids, using spheres instead of prisms. Tbe new method has several advantages over the older one: it is more accurate (two units in the sixth decimal. place is quite feasible), the trouble in carrying out the experiment is considerably diminished, and we use a theodoli te instead of a spectrometer. The last fact is of some importance because a spectrometer is about eight times as expensive as a theodolite of comparable quality.

A second application is based on the observation that the rain-bow remains perfectly at rest while the raindrops are falling. This leads to a new technique of focussing collimators and telescopes. With the aid of a glass sphere this adjustment can be carried out (for both instruments independently) within a quarter. of an hour; the accuracy attained approaches interferometric-precision.

The aberrations of the collimator lens (and of the objective of the telescope) can be fairly accurately measured, e. g. spherical-and chromatic aberrations, field curvature, astigmatism.

Moreover we can measure large radii of curvature. Radii exceeding one meter can easily be determined with a fairly great accuracy; in many cases the new method can replace the rather troublesome measurements with Foucault' s method. In testing the flatness of plane mirrors deviations as small as one quarterof a wavelength can be detected.

Other applications, e. g. the alignment of points on a straight line or in a plane, are only briefly mentioned. Purely technical details and industrial applications are left aaide; we only wanted to show some of the sphere's possibilities. Nor did we aim at covering the whole field of applications; in investigations like this completeness does not exist.

PART 1

GEOMETRICAL OPTICS OF THE SPHERE

1. I NTRODUCTION.

From the viewpoint of geometrical opties the solid glass sphere is a particularly simple example of an axially symmetrie system. We will choose an arbitrary straight line through the centre of the sphere as the axis and we will study the image formed by the sphere of a plane perpendicular to this axis. We will assume that the light between entering and leaving the sphere is subject to k internal refleCtions; k

=

0, 1, 2, etc.

Ray tracing in the usual way will not be necessary for a simple system like this; the formulae expressing the parameters that determine the emerging rays in terms of the parameters of the incident rays are easy to handle. Virtually those formulae contain everything that one might want to know about the sphere as an instrument for optical image formation.

It appears to be useful though to write the theory of spheres in a more conventional way as this will simplify the comparison with other optical instruments very much. Starting from the for-mulae mentioned above, we will derive the paraxial and the third order data of the sphere.

One might object that by confining ourselves to the third order aproximation our results will not be very rigorous , and that many an interesting detaU might escape our attention. But, as long as the aperture is not excessively large, this objection does not hold. The third order aberrations appear always to be large, and so the higher order aberrations would only slightly influence our results. This will be shown in a few cases where fifth order theory can be applied without too much trouble. We must admi1.' that the most interesting feature of the sphere: points of the caustic at infinity, gets lost. But this subject is extensively dealt with in the second part of this thesis.

The paraxial theory is treated with the aid of the matrix notation that is slowly getting into general use. [ 1; chapter IX b

J.

For the third order theory we will use the system devel-oped by Smith, Korringa, Stephan and Brouwer. [ 1, 2, 3, 4,]. Especially Smith's and Brouwer's TI-~-y- theory [ 5; not pu-blished

J

,

originally devised as a theory of thin systems, will also appear to be useful in the case of concentric systems.

2. FUNDAMENTAL RELATIONS.

The sphere is not only an axially symmetrie system, it even is a eoncentrie system and so every ray is a meridional ray; we need only trace the path of the rays in one plane that eontains the eentre of the sphere. Let, in fig. 1, C be the eentre of the sphere L: and let us take AC to be the axis. The sphere is sur-rounded by air, itl> refractive index is n .

.

,

I

,

"

/

A

' ,

_"

/

/

.

_._

.

_

.

_

.

_

.

_.*._

.

//C,

/ \. /

,

/

,

/

,

1. Path of a ray in a BoUd homogeneouB Bphere.

Let AB be a ray incident upon the sphere in the point B. If i is the angle of incidence and i' ls the angle of refraction, then the change in the direction of the ray in B will be (i - i'). We will measure this deviation clockwise.

The refraction in B is followed by kinternal reflections . For every refleetion the angle of incidence is i', and so every refleetion will add to the deviation of the ray an amount of ( n -2i').

Finally the light will be refraeted out of the sphere, and again the deviation due to this refraction is (i - t'). So the tota! devia-tion [l of the ray amounts to:

[l = (i - i') + k (n - 2 i') + (i - i') •

The incident rays are defined by the angle V with the axis and thelr distance h from the centre of the sphere. For the emerging rays we introduce quantities of the same kind: ":1" and h'. It is a direct consequence of SneU's law that h' equals h, and so the emerging rays are given by the equations:

V'

=

V - q, (2)

and h'

=

h . (3)

The angle q, must oe taken from equation (1), in which i and i' are easily found: i is given by

sin i

=

R. h (4)

R being the curvature of the sphere, and i' follows from SneU 's law:

n sin i'

=

sin i . (5)

If we introduce a cartesian coordinate system (x, z) with the origin in C and the z-axis along the axis of our system, the equation of the incident rays will be:

- z sin ":1' + x cos V = h (6) and the equation of the emerging rays:

- z sin (V - q, ) + x cos ( 'Y - q, )

=

h (7) The deviation q, is only a function of h, it does not depend on ":1'. It is worthwhile noticing that the simple cases of a concave or convex spherical mirror mayalso be treated with the aid of the formulae given above; a concave mirror is obtained by putting k

=

land n

=

1, and a convex mirror by putting k = 1 and choosing n infinitely large. In the latter case i' is equal to zero and q, = 2i + 11:, the value that q, should have in the case of a convex mirror.

At this point it is necessary to introduce a sign convention that allows us to apply the formules (1) ... (7) in any case. The angles i and i' are taken to oe positive when the rotation of the ray, af ter a refraction or reflection, towards the direction of the normal, over an angle smaller than 900, is clockwise. When it is anti-clockwise i and i' are· negative. ,

The sign of h is taken from equation (4), in which R is always positive; so i, i' and h always have the same sign.

Next the sign of V is taken so as to be in accordance with equation (6). With these precautions the equations (1) and (7) may always be applied.

If we use the x-axis as a reference axis both in the object space and in the image space, the expression for the numerical value of the angular eikonal function, [2, 6

J

,

becomes quite simpie. This function E is defined as the optie al distanee of the feet of the perpendiculars from C upon a ray in the object space and the corresponding ray in the image space. A moment' s consideration of fig. 1 shows, that

E

=

+

i {-

cos i + (k + 1) n cos i'

i

(8) Unfortunately this is only the numerical value of the eikonal functionj to obtain the eikonal function in its usual form i and i' must be expressed in sin 'Y and sin 'l! '. This yields a formula that is rather complicatedj as we have other means at our dis-posal· for the investigation of the sphere's properties we will refrain from explicity stating it.

3. PARAXIAL OPTICS OF THE SPHERE.

In the paraxial approximation we assume that all angles that are found in our formulae are, apart from a multiple of n:, so small as to allow us to inter change sines and tangents and the angles themselves. This means that we only deal with rays that remain close to the axis. We assume that the light in the object space travels from the left to the rightj i. e. the light is incident upon the left side of the sphere. This involves that cos 'l! is positive; in our approximation its value is + 1.

The paraxial form of the formulae (4) and (5) is

i

=

R . h

=

n i' (9)

and consequently

<P = 2 (1 _ k + 1) R h + kn:.

n . (10)

So the equations (6) and (7) that represent the incident and the emerging rays reduce to

-zIJl + x=h (11)

and

t

k + l } k

- z 'l! - 2 (1 - - n - ) Rh + x = (-1) h . (12) It is a little awkward in paraxial opties to represent a lightray in terms of an equation in a cartesian coordinate system; it is more convenient to determine it by its slope dx/dz ('l! resp. 1JIe.') and its point of intersection witb. a reference axia pe~endi­ CUlar to the axis of the system. For this reference axis we choose the x-axis of our coordinate system, both in the object

space and in the image space. The points of intersection will be denoted by Xo resp. x' e' Using this notation we can write for

(11) and (12): k+l IJ! I

=

'!' - 2 ( 1 - - - ) R x 0) e o n k (-1) _{xe '} = Xo (13) (14)

We have taken the refractive index of the surrounding medium to

be 1. In the object space we can simply use this value + 1. But

in the image space a complication ris es: when the number of reflections is odd, the final refractive index should be negative.

Consequently the refractive index of the image space should be'

taken (_I)k.

It will be denoted by n' e; its counterpart in the object -space by

Do·

So finally our paraxial relations become:

'!' '

=

':I' - 2 ( 1; k+l) R (noxû) , (15)

e o n

(15a)

or written as arelation between matrices: k+l -2(1-

~)}

(n

':1':

)

o 0

(

n' x .

P

e')

=C

e e (16)

It seems appropiate to introduce as new variables the

coordinates Xo and x' e multiplied by the refractive index of the

space in which they are situated; we will denote them by

x

and

x' .

When this modification is also applied to the distances along the

axis, (z and Zl instead of zand z') we have arrived at the

"modified notation" . (Cf. Stewart [ 7, page 12 ] and Brouwer

[ 4, page 24 ] .

In the "not modified notation" lengths perpendicular to the

axis are left unchanged; but the directions IJ! and '!' I are multiplied

by the proper refractive index, and consequently distances along the axis are divided by a refractlve index. This notation is also useful in some cases, e. g. the elementary treatment of thin

systems, but in aberration th~ry the modified notation has much

in advance, even more so in the theory of concentric systems as for concentric systems the modified power is additive, contrary to the power in the usual sense. So we will confine ourselves to the modified notation.

According to (16) the (modified) power J of the system is given by

J = 2 R (1 -

~)

(17)

It is obvious that the nodal points of the system coincide with the centre of the sphere. In order to find the other cardinal points we shift the reference axis in the object space and in the image space over an amount of zand "2' respectively.

The matrix with respect to the new axis is given by

(1 0) (1

_Z'

_{1 0 1}

-J)

(1_

_-z

₁

0)

or -J, _ ,..) ( :

-~)

(1

+

Jz

\ Jz"2' +"2' -"2 (18)

Tbe reference axes are imaged onto each other when D =0, or

l

=1...+

J

"2' Z (19)

a formula not to be wondered at. One should consider though, that the distanees zand z' are measured with respect to the centre of the sphere, i. e. with respect to the nodal points of the system. Usually the unit points are used for this purpose. From (19), the first and second principal foei are located at: Z

= z

=

-~

z'F

=

(_I)k z'

= +.1

(20)

F F J ' F J

80 the two foei coincide when the number' of reflections is odd, a fact quite obvious when we observe that the direction of a lightray may always be reversed.

When in (18) D is equal to zero, G and B are the modified lateral magnification and the angular magnification respectively. The modified lateral magnification is the ratio

x'

Ix,

contrary to the ordinary lateral magnification ~'

= x'

Ix.

We see, that

(21)

The unit points are defined by ~' = + 1. When the number of reflections is even, they coincide with the nodal points; whell k is odd we obtain

2

z ' = - z ' _{u u}

=.!

J (22)

J

Tbose two points coincide too, a matter of course again when we consider the symmetry of the system.

Observing that the determinant of (18) always has the value + 1 we arrive at an important lemma: the product of the angular and lateral (not modified) magnification is equal to the ratio of the

refractive indices in the object space and the image space. This

lemma holds for any axially symmetric optical system. It enables

us to explain why in aberration theory the modified notation is more convenient than the ordinary notation. The point at issue lies in the formulae that are used in the addition of the aberra-tions of co-axial systems. These formulae are of paramount importance, as the aberrations for a combined system are always

obtained by adding the aberrations for its components, i. e. by

addition either per surface or per thin lens. (The latter method is falling into disuse because it puts too severe a limitation to the cases in which it can be applied). To obtain the contribution of one surface to the aberrations for the system, the aberration coefficients of this surface must be multiplied by a certain factor. This factor consists of the proper powers of the lateral magnification of the object plane and the pupil plane, both in passing through the part of the system following on the surface

that we consider. Those magnifications might be calculated by a

repeated application of the lens formula, but this is rather a

tiresome procedure. A good deal of time is found to be saved, when we paraxially trace two rays through the system: one ray passing through the axial point of the object, and one through the axial point of the entrance pupil. We choose those rays so

that their direction in the image space, . 'l" cq 'JIèp' is equal

to + 1. When this is done, the lemma mentioned above is used.

Fro~ this lemma we infer that the directions of the rays

'l:'

k

cq 'f'kD af ter refraction (or reflection, as the case may be) oy

the kth surface are numerically equal 1.0 the magnifications we

need in our addition formulae, provided we use the modified

notation: in this notation the ratio of two refractive indices, that one would expect to appear in our formulae, is absorbed into the definition of the modified lateral magnification. In the ordinary notation this simplification does not take place, and so the addition formulae become unnecessarily complicated.

When we only consider the position of the cardinal points, the sphere with an even number of reflections is equivalent to a thin lens with the power J. When the number of reflections is odd,

it is equivalent to a simple spherical mirror, with the centre

of curvature at C. lts curvature Rl is given by

R =

1

J n - k - 1 R (23)

1 2 n .

When J is negative, the concave part of the mirror is used; for

i. e. when our system is telescopic, this representation fails. This happens when e. g. k

=

1 and n

=

2. (And this is practically the only possibility). Such a system can be represented by a cornercube, the point of intersection of its planes being located at C.

All this only holds for the paraxial approximation. When aberrations are also taken into account these "equivalent systems" behave quite differently. For the monochromatic aberrations ·this will be shown in a subsequent chapter.

The chromatic aberrations of the sphere can calculated. The nodal points are always located at C only have to determine the change of the power J when the wavelength of the light. Thus, from (17):

6. J = 2 R k

:2

1 6. n

easily be and so we we change

When we write p = k + 1, and introduce Abbe's reciprocal dispersive power v, we have

tJ. J _ P (n - 1) r - n ( n - p )

1

v (24)

The consequences of this equation might be illustrated by com-paring it to the formula that holds for a single thin lens:

~= 1

J

v

We see then, that t;be sphere has a smaller chromatic aberration than a single thin lens when

I

p (n - 1)

I

<

n (n _ p) 1 , i. e.

n

>

p

+

yp

(p - 1) (25) Because of the limitations we have to put to the value of n this requires that p shall be 1, i. e. no internal reflections.

In that case:

/':,J

T

n 1

1

v (26")

80 the chromatic aberrations of a sphere without internal reflec-tions is considerably smaller than that of a single thin lens of the same power.

4. THIRD ORDER ABERRATIONS. (a) Int r 0 d u c t ion.

The theory of third order aberrations that we are going to use in this thesis is based on T. Smith's fundamental paper: The changes in aberrations when the object and stop are moved

(Trans. Opt. Soc. 23, 311, 1921/22). Smith's work has been

developed by Korringa (thesis, Delft, 1942) and Stephan (thesis, Delft, 1947), and it has become an extremely useful tooI in lens design. Some of the results have been summarized and derived in a simplified way by Brouwer (thesis, Delft, 1957). Brouwer and Beernink have studied the special case of thin systems. We wish to express our gratitude to them for putting at our disposal their most recent, as yet not published, results, because their conceptions will serve as a model for a theory ot concentric systems that will prove very useful for the description of the sphere 's properties .

In what follows the most essential points of the third order theory are summarized. We will use the modified notation throughout and we only deal with meridional rays.

We consider an axially symmetrie system of (modified!) power J. The object magnification is G, the pupil magnification S. A ray is defined by its point of interseétion with the object plane and the plane of the entrance pupil,

x

and

Xp

respectively.

This ray intersects the image plane and the exit pupil plane in points given by X' 'and X'p.

The deviation from the paraxial imagery is given by:

We introduce: and we write: i' -

Gx

.

and SXpJ S-G (27) i:'- G X = !(m1 1; 3 + m2 1; 21;p+m31; I;p2+ m 4I; p3).(28) Here mI ... m4 are the third order (or in Smith's language

primary) aberration coefficients for distortion, meridional

field-curvature, coma and spherical aberration respectively. Moreover we introduce mo in such a way that

represents the spherical aberration of the exit pupil, i. e. x~ as

a function of

x

for Xl = O.

The m-coefficients have the dimension of a length; E, and ';p are

of dimension 0; they represent the field angle and the aperture

angle in the image space.

In order to calculate mo ... m4 we paraxially trace two rays, passing through the axial point of the object and the entrance

pupil respectively. Their direction in the image space be + 1;

this involves that their direction in the object space is + G and

+ S respecti vely. (For G = 0 or 8 = 0 an~ obvious limiting

procedure yields that the ray shall be parallel to the axis, with a height of incidence -l/nJ). The ray tracing is performed

according to the following scheme (it is given for two consecutive

surfaces):

n : refr. indo obj. space

_I1.

n' : refr. indo image space n!

1

R curvature _~

t' distance to the next

surface

~

h : height of incidence _{~ hi + 1 = ~ + t'i 'l'}

_i

'!' : inclow. resp. to the 'f_{i 'i'i + 1 = '!'i'}

axis before refr. _~hi

+

cp : angle of incidence _{CPi ='l'i + Ri~}

cp' : angle of refraction _{CP'i = (~/~ )CPi}

d : deviation d = _{CPi - CPi'}

'1" : incl. W. resp. to the

axis af ter refr. _{'1' i' = '!'1 - di}

e : auxiliary . quantity e ='l'i +CPi'

P : auxiliary quantity _{Pi = ~~'/(~'-~)}

This scheme must be worked out for both rays; the quantities

related to the "S-ray", the "pupil-ray", are marked with an

index P. The scheme used is what Smith's ray tracing formulae

(Proc. of the Opt. Conv. 1926, II page 753, London) are reduced

to in the paraxial approximation.

Now mo ... m4 are calculated as follows:

mo = 1: (hp dp2

~

Ph

mI = L: (hp ddp

ep

P)ï + 8'jG ( 1 - 82) (29)

m2

=

-

3L: _{(hdp 2 e Ph - ( 8}

J

G)2 L:, (R/P}f

m3

=

3 L: (hddp e Ph

The meridional -and sagital curvature Rm and Rs are given by:

, J 2

R _m = n ( - - ) _S-G m2

Rs = 1/3 Rm -2/3 nt ~ (R/P)i (30)

in which L (R/p) i is the Petzval sumo

For practical purposes it is very important what happens to the

aberration coefficients when G and S are changed. We introd:uce:

Do = m 0

+;

(S-G)

f

_

-

~}

,

Dl

=

m1 1 D2

= -

3'

m2 (31) 1 D3

=3

m3 D4

= -

m4

When we denote the quantities af ter shifting the object and the

stop by overstriping them, we have:

4-i i {-4-i- i

1

Di=

(Do ...

D4 t 1 + s, - s) (g, l-g)

+}

(G-G) S GG G 1), (32)

G-G S-S

g

=

S -G s

=

S-G (33)

4-i i 4-i i

(Do .. ,D4 (a, b) (c, d) means: calculate (a+bt) , (c+dt) and

substitute

Di

for ti,

This is not quite as complicated as it seems to bei in many

cases (32) reduces to simple relations. One example: when only

S is changed we have for the coma coefficient:

na

= (

1 + s) D 3 - s D4

(34)

i. e. m3 = (1 + s) m3 + 3 s m4

When two coaxial systems A and B are combined, the addition

formulae for the aberrations read:

DiAB

=

DtA

~ S~-i

+ DiB (35)

Also for aspheric surfaces and chromatic aberrations simple formulae like (29) exist, but they are not essential for this thesis and we will refrain from stating them.

4. (b) Third order aberrations of concentric systems.

In this section we will prove that when the power J of a concentric system is known, the third order aberrations for any value of S and Gare completely determined by just one quantity Y. When this proof has been established, we only have to calculate

Y for the sphere in order to know aU its third order aberrations; this will be done in the next section.

When the stop is located at the centre of the concentric system, every ray passing through the axial point of the stop is incident perpendicularly upon every reflecting or refracting surface of the system. Consequently dip in the equations (29) is zero for aU i. 8 equals + 1 in this particular case and so

(29) reduces to:

mI

=

m3

=

0, l-G 2

- (-J-) ~ (Nt/Pi)

(36)

The formula for m4 cannot be simplified.

Ri /pi is the modified power of the ith surface. As we have mentioned before, this power is additive in the case of a con-centric system.

[

7,

page 12]. 80 the Petzval sum 2: Ri/Pi is equal to the power J of the system, whence

m2

= -

~

(37)

From (30) we see, that the meridional and sagittal curvatures both are -n' J; i. e. in the case S = 1 there is only field curvature and no astigmatism. We further specialize to G = -1, and we change from mi to

Di

according to (31):

Do

_J

4 8=+1 Dl 0 G=-1 D2 ₌

_3J

4 (38) D3 0 D4 _J2

.1.

y 20

The last equation is only the definition of "(. We see, that J

and 'Y completely determine the aberrations in the case of G

=

-1 and 8

=

+1. But, according to (32), this has as a consequence that J and 'Y determine the aberrations for any value of G and 8 . The quantities g and s occurring in (32) are in this case:

1

g

="2

(G + 1), s = 2' 1 (8 - 1),

and so we obtain:

.IDï.

=

!

(1, o,î,o, Y/J

l

1 + 8, 1 - 8) 4-i (1 +G, 1 - G)i

- (1 + G) (1 +

é-

iGi-1).

From this relation the coefficients Di can be calculated for arbitrary values of 8 and G. It can be reduced to a rather more attractive set of formulae by putting

Then we have: Moreover for we have M

=

1 + G V

=

1 + 8 (39) l - G 1-8 mI =

(1-~)Jl-S)3

[Y IJ

-

V(2M-V) ] m2 = _3(I-G):.?-8)2

[Y

/

J _

~

(2M2 + 2MV _ V2) ]. m3 =

3(I-Gl~(1-8) [Y

I

J -

M2 ] ' (40) m4 = -

(~JG)4

[

Y

IJ

-

M2 ] 8-G 8 3 mo

=

Do - - J - (1 -

G )

m 0

=

(l

i

ff)4 (Y

IJ -

V2 ) (41)

For 8 = +1 this indeed reduces to (36).

From (3~ and (40) we may draw the conclusion, that only one of the five third order aberrations can be chosen at wiW This one aberration coefficient determines the other ones,

vided of course that the power, and the positionof the ·object and stop, are known. Considering that the system may consist of any number of refracting or reflecting surfaces , this is a remarkable fact; it is mentioned bye. g. 8tephan [8, pag. 88 ] . It might be worth while to recall that "thin" systems have a similar property; theoretically they have three degrees of freedom for the aberration coefficients, practically however there are only two, as the Petzval sum is always roughly equal to seventy per cent of the power. 8mith' s theory of thin systems is based on this peculiar fact; Brouwer and Beernink have used 8mith's work to develop a routine for the straight forward design of thin systems [9]. The results for concentric systems that we have obtained above are modelled after their formulae.

We see, that the vanishing of the spherical aberration always involves the absence of coma, regardless the value of 8. This fact, at first sight rather striking, can easily be explained by equ. (34). It shows that the simultaneous vanishing of m3 and m4, if it occurs for any value of 8 at all, is independent of the position of the stop. But a value of 8 for which m3 vanishes, viz. 8

=

+ 1, does always exist.

Another interesting property of concentric systems may at once be derived from (40): when the spherical aberration is absent for the object magnification G, it also vanishes for the magnification liG. This is a special case of a theorem due to Boegehold and Herzberger. [ 10

J,

[

11

]

.

It is clear that the relations (40) and (41) are quite useful in the design of concentric systems, because they describe its possibilities, and even more important, its impossibilities.

Incidentally it may be remarked, that to the lens designer the third order theory of concentric systems is not quite as important as the algebraic theory of th in systems. Designing thin systems by ray tracing is an awkward procedure; when one of the constructional data of the system is changed the ray tracing must be repeated for the part of the system following on the element that has been altered. In concentric systems however the surfaces are independent of each other, at least when the ray tracing is performed goniometrically. For more details on this question we refer to Hekker. [10].

For the moment we are left with one problem: how do we calculate the value of y for a system when the constructional data are given? y represents, to a factor -4/

J2,

the spherical aberration for an object such that the modified lateral magnifica-tion . is minus one. 80 we might apply the method indicated at page 18. It is however advantageous to have a more direct way of finding y.

To arrive -at a more convenient formula we must take recourse to the following relation holding for one surface:

Q denoting nn' / (n' _n)2, an equation that can easily be checked by applying (29) to one surface. From this equation we can determine D4 for any ~value of G, i. e. for any position of the object. The stop does not enter into the equation as D4 is independent of the stop's position.

If we knew the partial magnifications of the surfaces in the system we are considering, we could first apply equ. (42) and then the addition formula (35).

Those magnifications are easily found. When we add two systems A and B, the following paraxial relation holds:

GB JAB

=

JA + GAB JB . (43) When the object is at infinity GAB = 0 and so

_ JA

G_B - JAB ₍₄₄₎

Thus if we add the first i surfaces of the system to the rest of it, we obtain:

(45) The system being concentric Jl, i is simply equal to the sum of the powers of the first i surfaces . A repeated application of

(42) and (35 yields:

m4

=

~ ~i

(Gi+1,n - Gi , n)2

l(Gt+1, n - Gi , n)2

~

- G_{i ,} n Gi +1 , nJ For G = 0 (45) can be substituted into this equation, so that

m4/

~

2:

i J i

{Jt

Qi - J 1, i-1 J 1,

d .

(46)

G = 0 c.I'

Thè chaqge to G

= -1 is most easily done by applying (40); the

result being:

4 2: , 2 }

Y

= J + J2

1 Ji lJi ~ - J1,i-l J 1 ,i (47)

Owing to Boegehold and Herberger' s theorem mentioned

this is equal to

Y

=

J +..L _J2

~

₁ Ji 1.r._{l 1 Qi - Ji ,n} J. 1 _{1+ , n S ,} 1 which can be verified algebraically *).

above

(47a)

*)These formulae can still be considerably simplified by observing that

y J2 +

j

J3 is additive. (note added in proof).

4. (c) Third order aberrations of the sphere. Once we know

r

,

the equations (39) and (40) completely

describe the third order properties of the sphere. To determine Y we only have to insert the appropriate values of Ji in equ. (47).

The first and last deviation of the ray are caused by refraction. The powers entering here are:

Jrefr n-1 _n R

and for both refractions the value of Q is

Q

=

_(n-1)2 n

In between entering and leaving the sphere the light is subject to k reflections, the power for each reflection being:

2R

Jrefl = - 1 1

For a reflection Q is equal to -1/4, independent of the refrac-tive index of the medium in which the reflection takes place.

Substituting all this in equ. (47) we obtain, af ter a rather lengthy computation:

2 Y/J =3np(n-p)+p(p -1)

3 (n _p)3 (48)

in which p is equal to the number k of internal reflections , plus 1. A few conclusions may be drawn from this formula.

When Y is equal to zero, the spherical aberrationis corrected for G = -I, i. e. a symmetrical image formation. When we put the numerator in (48) equal to zero we see, that p should not be greater than 2. Fbr p =:= I, i. e. no internal reflections , we arrive at n

= 0, which is physically impossible, and n

= 1.

The last solution actually does not satisfy the equatiQn y = 0; more-over. it means that rays pass on without being subject to refrac-tions at all, the system is nonexistent.

For p

= 2 we arrive at n

= 1.

This solution means, that the centre of a concave spherical mirror is imaged sharply on itself; a rather trivial conclusion. The same is found to be true for a convex mirror by choosing the refractive index infinitely large.

Correction of the spherical aberration is only possible at all when y

IJ

is non negative. In the case that there are no internal

reflections (p 1) this actually obtains:

riJ

=

(n~1)2

(49)

For p not equal to one the numerator of (48) is non negative definite; so correction for particular values of G is only possible when n is greater than p. Thu~ for systems with one reflection n should be greater than 2; which is hardly feasible. More reflec-tions seem to be out of the question when correction for spherical aberration is wanted.

The magnifications for which a sphere without internal reflec-tions is "spherically corrected" are readily found from (49), (40) and (39). They dep end on the refractive index;

n G 1,2 0,691 1,4 0,495 1,6 0,356 1,8 0,253 2,0 0,172

or the reciprocals of those G-values. This result seems not to be of great practical importance. To obtain correction for an object at infinity the refractive index should be 2,618. This can only be realised when certain minerals are used, f. i. rutile; the focal points are virtual in this case.

When n equals p the system is telescopic. It is possible to apply the theory developed above to telescopic systems; only, however, af ter a limiting procedure. As long as the object and the stop are not located at infinity, we need not change the definition of the aberration coefficients m 1 ... m4; The equations that define I; and I;p (page 10) cannot be maintained, but their meaning, the field angle and the aperture angle, need not be changed.

The positions of the object and stop have up to now been determined by their magnifications G and 8. This bas no sense here, as the magnification by a telescopic system is independent of their position. 80 we have to return to more conventional variables : the modified dis tance

z

measured along the axis. The centre of the system is taken to be the origin. From elementary paraxial opties we lmow that G

= 1 -

Ji'. This must be inserted into equation (40), which determines m4 as a function of J, y and G. J and

r

can be expressed in n, pand R, according to (17) and (48); this is substituted in (40) too. When this is done the I.:.mit for n approaching to p can be determined. The result proves to be:

(50)

When the object moves to infinity the deviation from the perfect image formation must be expressed in terms of angles instead of lengths. Eventually we find:

Óx'

.

1 3

lim - - - =-ÓL' = - R

z'-=

z'

3

(51)

Hekker [ 11, page 51 ] states a theorem that concentric systems of zero power are always perfectly corrected for any position of the object; it is evident from the results obtained above that this theorem does not hold in general.

Summarizing the results of this section we see, that except for a few cases of less importance the sphere with one or more reflections is badly affected with spherical aberration. The third order spherical aberration of a sphere used as a thick lens without internal reflections vanishes for certain positions of the object, either the object or the image being virtual.

This result seems to be rather disappointing; apparently the sphere has not the properties desired in an instrument used for optical image formation. In a subsequent chapter we will see however that certain other applications of the sphere entirely depend on its very large spherical aberration. For tbose applica-tions it will appear- necessary not to restrict ourselves to third or fifth order aberration theory; the path of the rays must be calculated rigorously. Moreover the fifth order theory is only of interest when the third order aberrations are small. So we will refrain from completing this chapter with a 'discussion of the fifth order image errors.

First however we will devote a few more pages to the sphere used as a lens. It seems to be of at least some academic interest to find out what are the properties of the spherical single lens compared to other thick single lenses. We will only study a case that once was currently employed in practice: no reflections and the object at infinity. The application we have in mind is the use as a magnifying glass,notably Van Leeuwenhoek's microscope. Actually we should take the image to be at infinity; for the calculations however it is convenient to reverse the direction of the rays.

5. THE SPHERE AS A MEMBER OF THE CLASS OF TRICK SINGLE LENS ES.

The lenses used by Van Leeuwenhoek in hls "micr08cope" (we would rather use the word magnifying glass) were almost,

but not quite spherical bulbs of glass. [14

J.

One has often won-dered why he did not use glass spheres, as, for one thing, the manufacturing of spheres is considerably easier than making lens es like Van Leeuwenhoek used them. This is not only true when the spheres are made by melting glass globules, [ 12,

Vol. 3, page 55

J,

they cao also be manufactured easily by grinding and polishing .

Moreover it is generally believed that the spherical shape is the very best shape as far as the aberrations- are concerned. In

this section we will try to find out whether this is true. We will consider the class of thick single lens es of power one. The object is supposed to be at infinity (we reverse the direction of the rays merely to simplify the calculations ) and we will calculate the third order spherical aberration of those lens es . This is sufficient; the spherical aberration is the only aberration of interest in this case, and, as long as the aperture is not excessively large, the higher order aberrations play no part as the third order image errors appear always to be great.

A sinJle thick lens is characterised by four parameters: the curcatures RA and RB of the first and second surface, the thickness t, and the refractive index n. Instead of the curvatures RA and RB we will rather use the modified powers JA and JB in our formulae, and on the occasion we will write for them x and y: n -1 R x = - n - _A n - l y - - - -_n RB (52)

The power of the lens is supposed to be unity; this requirement enables us to easily compare the various lenses that we will consider. It involves a relation connecting the four data that determine the lens:

1

x + y - t x Y = n (53)

The calculation of the spherical aberration coefficient in terms of x, Y. tand n goes along lines similar to the derivation of y on page 22. The partial magnification of the first surface is obviously zero; that of the second surface follows from:

In our case we have J_AB

=

1 and GAB

=

0, so that

(54)

(42) gives us the spherical aberration of one surface; for the .

combination of the two surf aces we have the addition formulae (35).

The result is:

3 (1-x)2! (1-x)2Q-x}

D4 = Qx + Y ₍₅₅₎

or

D / 4Q-x - 3 + (x-1)2 (x-n)(x--!:.) n

y (56)

Q denoting n/(n-1)2. Instead of y we can introduce t as a variabie; (53) yields:

D4/Q = x3 - (x_1)2 (x-n)(l-tx). (57)

For the sphere we have: x

=

y

=

!,

so its 3d order spherical

aberration coefficient becomes

D4

=

!

(Q - 1), _{. (58)}

as also follows from (40) and (49).

First we will consider the subc1ass of concentric lenses. These systems are characterised by the sum of the modified

powers JA and JB being equal to + 1:

x+y=1. (59)

When this is substituted in (55) we eventually arrive at:

D4 =

!

(3Q + 1)(x_y)2 +

!

(Q-1). ₍₆₀₎

Those expressions are symmetric in x and y, a direct

con-sequence of the considerations on page 22. When the refractive index is kept constant the minimum is attained when x and y are

equal, i. e. when the lens is spherical! 80 in the subclass of

concentric lenses the sphere bas the least third order spherical aberration.

Another subc1ass containing the sphere is composed of the symmetric lens es ; those lens es are characterised by x and y

being equal. When we replace y by x in equ. (55) and differentiate with respect to ,x, we find by putting the result equal to zero an equation for the value of x that will minimize D4 for a fixed value of the refractive index:

4 2 3 2 2

6 x - (8 + Q ) x + (6 -

Q )

x - 1 = O. (61) This equation has one positive root, and this is the only root with a physical significance*). When this value of x is computed, equ. (53) yields the thickness of the system.

The results for various values of nare collected in table I, together with some other data: 1. the ratio of the thickness and the radius of curvature of the refracting surfaces , 2. the obtained minimum value of D4, and ~. the value of

Di

for a sphere with the same refractive index. We see, that the best shape of a symmetric lens is almost spherical; it is spherical indeed wh en the refractive index is 2. For practical purposes the difference can safely be neglected.

TABLE I n x tir D4 D4 sphere 1,3 0,5343 2,4280 3, 257() 3,3611 1,4 5306 2,2882 1,8847 1,9375 1,5 5263 2,1997 1,2243 1,2500 1,6 5217 2,1387 0,8482 0,8611 1,7 5166 2,0917 0,6114 0,6173 1,8 5113 2,0554 0,4509 0,4531 1,9 5057 2,0250 0,3359 0,3364 2,0 5000 2)0000 0,2500 0,2500 2,1 4942 1,9786 O,1834x ) 0, 183SX) X): F' virtual.

Data on best symmetric lenses.

It might be of some interest to' compare these results to the subclass of thin lenses. For thin lenses we have the relation: x + y

=

1/n. When we insert t

=

0 in (57), we arrive at a simple expres sion for D4:

*)In order that the thickness shall be positive, x and y must not be smaller than I/2n. The relative maximum attained at this boundary is of.na interest.

D4

=

Q [(2 + n) x2 - (1 + 2n) x + n ] (62) Any value of x is allowed, and consequently the absolute minimum is attained for

1 + 2n

x

=

2(2 + n) (63)

its value being:

_ n (4n - 1d

D4 - 4 (n - 1) (n + 2) (64)

Again the best shape depends on the refractive index. The re-sult does not differ very much from the plano convex lens; the surface with the greater curvature is always turned toward the direction of the incident light. We actually arrive at a plano convex shape when the refractive index is 1,68614.

For a plano convex lens we have:

2 - n

D4

=

1 + n (n _ 1)2 (65)

Some data on those thin lens es are collected in table Il.

TABLE 11

n RB/RA D4 D4, plano convex

1,3 -0,4102 4,5960 6,9829 1,4 -0,2782 2,9596 2,6786 1,5 -0,1667 2,1428 2,3333 1,6 -0,0714 1,6667 1,6944 1,7 +0,0107 1,3596 1,3601 1,8 +0,0821 1,1470 1,1736 1,9. +0,1447 0,9924 1,0650 2,0 +0,2000 0,8750 1,0000 2,1 +0,2491 0,7831 0,9606

Data on best thin lens es.

The last two colums of table I and 11 are depicted in fig. 2. D4 changes rapidly with the refractive index; for small refractive indices the spherical aberration of a certain type of single lens is roughly proportional to 1/(n-1)2. We see that the spherical or approximately spherical shape bas great advantages over the thin lens.

As we have mentioned before, Van Leeuwenhoek did not use the sphere but a lens that was rather thinner; for the specimen

2. Third order spherical aberration of single lenses as a function of the refractive index.

A: Best symmetrical shape. (thick single lens!) B: Sphere.

C: Thin lens of best shape.

investigated by Van Cittert [14 ] the ratio of the thicknessand the radii of curvature was 1,467 instead of 2. When the refractive index is taken to be 1,5, this yields a value of 0,441 for the

~odified. pow.er of the surfaces, and a thickness of 1,108. D4

IS 1,522 in thlS case; twenty per cent greater than the spherical aberration of a sphere made of the same material - 80 the sphere appears to be a little better than Van Leeuwenhoek's lens. However, it would he quite wrong to draw the conclusion that the sphere (or the symmetrie lens of table I) is the best single lens for use as a microscope. For we have as yet only investi-gated three special cases: concentric, symmetrie and thin single lenses. To study the single lens more generally we have prepared fig. 3: lines of equal D4 in a x-t-diagram for a refractive index of 1,5. In this figure we have suppressed the combinations of t

and x that would produce a virtual focus: 1-xt must be positive. The diagram shows that it is possible indeed to make single lenses better than the sphere; a lens characterized by t

=

2,

x = 0,4 for instanee has a D4 as small as 0,8592; its curvatures

are 1,2 and -4. Fig. 3 shows quite clearly that neither the sphere, nor Van Leeuwenhoek's lens have any special properties compared to other thick single lenses, as far as the aberrations are concerned.

In fig. 3 we have reached the object of this section. It gives us a clear picture of the entire class of thick single lenses and the position of the sphere in this class. We will not go into further

details, the diagram speaks for itself. The values of y belonging to any combination of x and t can be drawn from fig. 4. For other refractive indices the results are similar .

Finally we would like to justify, at least for one special case, our statement that the higher order aberrations can safely be neglected, even when the relative aperture is fairly large.

We will demonstrate this hy calculating, as a function of the height of incidence, the later al spherical aberration in the paraxial image plane of a sphere with a refractive index of 1,5. The object plane is still at infinity; the power is assumed to be one again, so the height of incidence in the object space is roughly equal to the numerical aperture. In table 111 we have tabulated the third order approximation, the third-plus-fifth order approxi-mation and the results of ray tracing*).

Even for an aperture as great as 0,35 the discrepancy between ray tracing and third order approximation is a mere twenty per cent. Of course, when we increase the aperture to still greater values, the infuence of the higher order aberrations rapidly increases and cannot he neglected any more.

*)The fifth order spherical aberration of the sphere without reflections, the object being at infinity, is given by ~ M6 f..~ , in which M6 is equal to

1 2

w w

t

t

2L

I

1

15

20

JO

I

50

I

1.5

,

2

...,..

.

_. - symmetricol lenses

- - -- concentric lenses

x -axis:

t

hin

lense~

A

:

van Leeuwenhoek lens

8:

Sphere

n

=1.5

J =+1

-1

o

1

2

3. Thick single lenses: lines of constant D4 in a x-t-<liagram. x-t-combinations that yield a virtual back-focal-point have beer.

suppressed.

w

"'"

t

t

2

1

~~~/

Y

0.750

n=1.5

J=+1

0'

I " I , J " > , I

-1

0

1

2

f

i

TAB LE III

h sin 'f'~ 3 e orde 3e+5e orde streng

0,05 0,05008 0 ,00008 0,00008 0;00008 0,10 0,10063 0 ,00062 0,00063 0,00063 0,15 0,15216 0 ,00211 0,00218 0,00218 0,20 0,20520 0 ,00500 0,00529 0,00531 0,25 0,26040 0 ,00977 0,01066 0,01077 0,30 0,31850 0 ,01688 0,01908 0,01951 0,35 0,38046 0 ,02680 0,03160 0,03293

Most systems that are not correctedfor thirdorder aberrations behalve similarly. The point up to wbich the third order approxi-mation is sufficient depends on the accuracy required whlch depends largelyon the size of the system (see the next section), and also more or Ie ss on the construction of the systemj table

m

gives however a fairly good idea of what generally happens. 6. VAN LEEUWENHOEK'S MICROSCOPE.

The conclusion of the preceding section is, that the sphere and similar thick single lenses are badly affected with spherical aberration. And yet, Van Leeuwenhoek obtained wonderful results with bis simple microscopes : a magnification of 300 times and

a

I resolving power not far from one micron appeared to be quite

possible. This apparent contradiction requires an explanation. First of all we must take a cIos er look at the rather vague concept of "resolving power". In practice this quantity is deter-mined by observing periodic structures like diatomees or test plates, e. g. Nobert's test plate [12, Vol. 1, page 404]. So we need not talk about two point resolution, a concept wbich is very important in astronomyj for our purpose the frequency response of a system is essential.

Furthertrtore the way of illumination is important. We can use incoherent, coherent, or partially coherent illumination. We will discuss here only the completely coherent and the completely incoherent case.

The third point we have to hold in mind is the very small size of the leJlBes. When we reduce the size of an optical system by a factor 2, the aberrations from the wavefront are also reduced

to half their original value, and these wave aberrations are eventu-ally the only things we are interested in.

In the case of coherent illumination parallel to the axis the . lens is a perfect low pass filter (see f. i. [ 15] ). Every frequency up to a certain cut off frequency is transmitted without any loss of contrast. The cut off frequency is determined merely by the numeri cal aperture: its value is given by the numerical aperture

devided by the wavelengtll. (The dimension of this Zine frequency is a length to the power minus one). Aberrations do not deterio-rate the contrast of any frequency, lower than the cut off fre-quency, that we want to transmit; they induce however a spatial phase shift depending on the frequency. Consequently aberrations may serously damage the similarity of a not sinusoidal object and its image.

A simple and weIl known example of this phenomenon is the out of focus observation of phase objects. Defocussing introduces a spatial phase shift proportional to the square of the line frequency, and just because· of this phase shift the object, invisible when viewed in focus, gives an image that can be seen by the eye. [16, page 35J.

The resolving power however, if it js defined as the finest structure to be resolved by the instrument, is independent of the aberrations. This result does not satisfy our intuitive ideas on resolving power. Indeed it would be better to define a "figure of merit" depending on the similarity of the object and the image and on the statistics of the set of objects to be investigated. For incoherent systems such figures of merit have been introduced by Linfoot. [17

J

.

When the same instrument is used with incoherent illumina-tion, the ultimate cut off frequency is twice as great as in the coherent case, There is, however, a decrease in contrast depen-ding on the frequency; very coarse structures are transmitted with hardly any loss in contrast, but when the line frequency of the object increases the visibility in the image gradually decreases until we have arrived at the cut off frequency where it is reduced to zero. The shape of the visibility vs. line frequency graph dep.ends largelyon the aberrations; it appears that, if Rayleigh's quarter wavelength limit is not surpassed, the deviation from the state of perfect correction can be tolerated, but that larger aber-rations seriously impair the transmission of high frequencies.

Let us try to find out what the consequences of Rayleigh's limit are in the case of the Van Leeuwenhoek microscope.

When a unit power lens bas a third order spherical aberration coefficient m4, a lens with the same shape, whose power however is J, will have a spherical aberration m4/J instead of m4. (We recall that the coefficients mi that we have introduced in section 4a have the dimension of a length). 80 the lateral spherical aber-ration

Óx'

in the paraxial image plane, as a function of the height of incidenxe X. is given by:

Xl J ispractically equal to the numerical aperture u of the parti-cular zone determined by xl' (WeIl to be distinguished from the 36

numerical aperture in the customary sense which we will call the maximum numerical aperture). One integration yields the deviation from the wavefront N:

N

=.!..

m4

u4

8 J (66)

In general a slight shift of focus is introduced in order to reduce to a minimum the influence of the aberrations. This is represented by an additional term in (66), quadratic in u. The coefficient of this term can be chosen in such a way that the overall value of N is a minimum. As a criterion one might use for instanee the requirement that the maximum value of N as a Junction of u is as small as possible. The resuIt proves to be:

the maximum being:

(67)

In these formulae Uo represents the maximum aperture.

Wben other criteria are used to determine the shift of focus a slightly different numerieal resuIt is obtained, but those small differenees are not essentiaI. Wben we want the deviation from the wavefront to be smaller than a quarter of a wavelength, we arrive at:

0,021 mJ4

é

<..!.

À.

o 4

(68)

The magnifieation V of the lens, used as a magnifying glass, is equal to 250.J, when J is expressed in reeiproeal millimeters. Wben we take À. = 0,56.10-3 mm, we ean write for (67):

4 V

>

38300 m4uo

On the other hand there is no sense in inereasing the mag-nifieation to unduly high values. The size of the smallest strue-ture to be resolved is À./2u. Wben this detail is magnified so as

to be seen at an angle of four minutes of are, we have reaehed the upper limit of useful magnifieation. (Rather an arbitrary limit, 'we must admit, whieh is, however, gene rally used). This involves, that

Consequently when we use incoherent illumination the maximum useful magnification that can be obtained with a Van Leeuwenhoek microscope is

V = 290 m-1/ 3

max 4 (70)

When we take m4 equal to 1,25, areasonable value according to the preceding section, we arrive at a maximum magnification of 270 times. The appropriate numeri cal aperture is 0,27, and consequently the ultimate resolving power is about one micron.

A sphere of this type would have a diameter of 1,4 mmo

We see, that only the cube root of m4 enters into our final result. This means, that the ultimate resolving power does not dep end seriously on m4; it makes hardly any difference whether its value is 1,5 or 1,25, and so the question "Why did not Van Leeuwenhoek use a sphere instead of bis own lenses?" vanishes into nothingness; it does not really matter.

We must admit, that from the purely theoretical point of view greater resolving powers can be realized: larger apertures can be tolerated provided that the size of the lens is sufficiently smaIl; a certain amount of useless magnification, however, must then be taken into the bargain. When we want to increase the resolving power by a factor 2, the lens must be made 16 times as smaIl, and so the magnification must be 8 times as great as the value that it would have in an ordinary microscope with the same resolving power. And tbis process is limited first of all by the finite radius of the sphere (~) and secondly by the huge values the spherical aberration attains when we further increase the aperture : the third order aberrations cease to be a sufficient approximation. For practical purposes all this is not of any interest.

We did not discuss the chromatic aberrations; Van Cittert proved that they are sufficiently small. [14].

And what was the performance. of the actual Van Leeuwenhoek microscopes ? We are not very weIl informed on this question. The best specimen in the collection of microscopes that Van Leeuwenhoek bequeathed to the Royal Society had a 200 times magnifying power [18, page 319]. The microscope at present

in the possession of the Utrecht University was thoroughly inves-tigated by Van Cittert; he states that lts magnification is 270 times and lts resolving power close to one micron, the aperture being about 0,5. The large aperture suggests that Van Leeuwen-hoek used coherent illumination, wbich is quite possible; according to Dobell [18, p. 331 ] he even might have used dark ground

illumination~ The way in which this was achieved is not lmown however.

Scbierbeek asserts that Van Leeuwenhoek probably possessed a 480 power microscope [19]. It is known that even more 38

powerful microscopes have been manufactured; the Italian Pater Della Torre appears to have obtained a magnification of 2560

times [12, page 53]; but it is doubtful if there is any sense in

such excessively. great magniffications.

From Van Leeuwenhoek's observations it appears that about

one micron was the limit of his resolving power, aresult well in accordance with our calculations.

For further historica! details we refer to the rather extensive literature on this subject, e. g. Harting [12], Dobell [18] and