II Applications of Shearing Interferometry

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APPLICATIONS OF SHEARING INTERFEROMETRY BY

OLOF BRYNGDAHL

Institute of Optical Research, Royal Institate of Technology, Stockholm 70, Sweden

CONTENTS PAGE

5 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . 5 2 . EXAMINATION O F OPTICAL COMPONENTS . . . . 5 3 . STUDIES O F ILLUMINATION P R O P E R T I E S . . . . 4

. EXAMINATION O F LARGE TRANSPARENT OBJECTS

5 5. STUDIES O F PHYSICO-CHEMICAL PHENOMENA I N

...................... 9 6. EXAMINATION O F MICROSCOPIC OBJECTS . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . . . LIQUIDS

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66 71 81

1. Introduction The study of shearing interferometers and their applications is a subject which has been very much developed during recent years. Examinations performed by means of interferometry consist in making the variation of separation between two wavefronts visible and measurable. I n shearing interferometers two superimposed, displaced images of the wavefront under test are made to interfere with each other, i.e. they provide a comparison of a wavefront with a sheared image of itself. An advantage of such instruments is that the need for a reference surface is eliminated, so that they can be used, in principle, for testing wavefronts of any size. Typical of the shearing interferometers is that they work without reunion of waves. The deformation, which a divergent or plane wavefront has received on traversing an object, is here made visible by interference in the following way: the deformed wavefront is divided by means of a beam splitter into two similar wavefronts, which are inclined at a small angle or are laterally displaced with respect t o each other. The essential equipment for this shearing is a beam divider, which may be any optical device that splits each ray passing through it into two rays. This may be performed in numerous ways, as will be seen later on in this article. The difference and simplification in respect to usual interferometer types are that here only one beamdividing element is necessary. The wavefront shearing interferometers may be used to test any convergent or plane wavefront, and therefore they have found practical application in many fields.

2. Examination of Optical Components 2.1. PRELIMINARY REMARKS

Wavefront shearing interferometry can be used for testing optical instruments and components, for determining performance and 39

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wavefront form. The shape of the wavefront is a function of the optical elements (lenses, mirrors, gratings, etc.) producing it. Consequently, the aberrations of lenses and the shapes of mirror surfaces are obtainable from the measurement of wavefront shapes. The image formation by an optical system may be treated by means of optical transfer theory. A principal feature that has come from it, is the possibility of evaluating the transfer curves, and therefore the image quality, from the constructional data of the optical system, The wavefront shearing interferometer may be used to study the transfer characteristic of any optical system in terms of its contrast response as a function of spatial frequency. Most usually the wavefront is characterized by the shape of the optical element or combination of elements producing it; thus, the evaluation of the wavefront is indirectly an evaluation of the element. The wavefront shearing interferometer can be used to measure the shapes of reflecting surfaces and aberrations of lenses. As far as the author is aware shearing interferometry was applied for the first time by WAETZMANN [1912] to test optical systems. He used a plane parallel glass plate as beam splitter, but, owing to inconvenient geometry of the optical setup (a modified Jamin interferometer), no quantitative measurements from the interferograms were performed (cf. HARTIIOLOMEYCZYK [1962]). RONCHI[ 1923, 19261 developed the well-known wavefront shearing principle for general optical testing. A transparent diffraction grating was employed as beam divider. It is very effective except that it provides more than two divergent wavefronts. In order to avoid the overlapping of more than two displaced images of the object under test, the width of the test field in the direction of shear must not exceed the amount of shear. 2.2. 'LESTTNG OF OBJECTIVES AND MIRRORS

2.2.1. Determination of the shape of wavefronts

A unique method of analyzing wavefront shearing interferometer data has been given by SAUNUERS [1961]. Hc used mathematical operations which do not require the assumption of revolution symmetry, or any other a priori information about the wavefront. The shape of the unknown wavefront is here given by its deviations from a chosen fictitious reference surface. This treatment is applicable to three dimensional space, but for simplification it will be limited to two dimensions here. By this method any convergent wavefront may

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be tested along any chosen diameter or along the line of its intersection with any plane that passes through the point of convergence. A number of equally spaced reference points are chosen along the above mentioned line, which must be parallel to the direction of shear. The separation of the points must be an integral multiple of the shear.

(c) Fig. 2.1. Illustration of analysis of a wavefront shearing interferogram: (a) Two images of a wavefront sheared laterally relative to itself; (b) Representation of the two wavefronts relative to the images of a reference circle; (c) Relationship between parameters and other variables.

In Figs. 2.1.a and b, P, represents the reference points in one of the images W of the sheared wavefront and P', the corresponding points in the other image W'. The separation of the reference points will be taken as unity. If the magnitude of shear has the same amount, one fringe pattern is sufficient t o evaluate the deviations of the wavefront at the reference points from a chosen fictitious surface. For another shear value more adjustments of the system are necessary. We introduce the following symbols:

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deviation of the wavefront at P, from a reference circle C (to be chosen later); 6, is positive if P, is on the concave side of the reference circle and negative if it Iies on the convex side, E = angle between the two images of the reference circle at their point of intersection, p = distance from the intersection point of the circles to the point corresponding to Po on C; p is positive if Po is below (cf. Figs. 2.l.b and c) the intersection of the two circles and negative if above, Y = distance from Po to P, measured along the circle, S, = separation of the two images of the reference circle a t any pair (assuming unit shear), of points P, and PVml Q , = separation of the two wavefronts W and W' at P,, 2h = distance between the centers of the two reference circles, R = radii of the circles, 4 = angle subtended by ( v - p ) at the middle point of the distance 2h (cf. Fig. 2.l.c), p = distance from the middle point of the distance 2h to the point corresponding to P, on the reference circle (cf. Fig. 2.l.c). All quantities that represent optical distances are given in units of the wavelength used. From Fig. 2.1.b we have 6,

y =

6,

=

S,-l+Q,--S,,

(Y =

1, 2 , 3 . . . N ) .

(2.1)

Applying the cosine law to the shaded triangles in Fig. 2.l.c, we obtain R' = /I"+ (p+S,)2-2h(p+S,) cos (90"-4) and R2 = h2+p2--2hp cos (goo+$), which give S , = 2h sin 4. (2.2) This expression may be applied rigorously, but with sufficient accuracy sin CP may be replaced by (v+p)/R. Also, since the angle E will always be quite small, 2k can be replaced by RE.Insertion of these expressions in eq. ( 2 . 2 ) gives

s, = (Y+P)&.

(2.3)

Then eq. (2.1) becomes

S, = 6 , - 1 + Q , - ~ ( ~ f p ) ,

(Y

= 1, 2, 3

. . .N ) .

(2.4)

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If Q, represents the unknown order of interference a t an arbitrarily chosen point P, (see Fig. 2.l.a), and qy the difference in number of fringes between this point and P,, then the order of interference Q, at P, is equal to Q,+q,. We can now introduce this and replace ep with a new parameter Y , so that ep = Q,-r, in eq. (2.4),which then becomes 6,

=

b,-,+q,+r-ve,

(v = 1, 2, 3 .

. .N ) .

(2.5) Equation ( 2 . 5 ) represents N equations with ( N + 3) unknowns: (N+l) 6,’s and the two parameters E and r. For a solution to be possible we need three additi,onal equations relating these ( N + 3) unknowns. The three additional equations needed for a complete solution may be introduced by defining, for example a reference circle. O f - course, the S,’s do not have significance before the reference circle C (see Fig. 2.1.b) is defined. The shape of the unknown wavefront can now be obtained by the amount by which it deviates from a mathematical circle (sphere) that is adjusted t o fit the wavefront according to any of the statistical methods used for fitting analytical formulas to empirical data. I n the article by SAUNDERS [1961] three principles, namely, the method of coincidence, the method of averages, and the method of least squares have been applied and discussed. 2.2.2. Measurement of wave aberrations

Shearing interferometers can also be used to obtain the various aberration coefficients of an optical system which are used in the [1950]. classification of wavefront aberrations as given by HOPKINS Here one utilizes the coordinates shown in Fig. 2 . 2 . AO, is the optical I

Fig. 2.2. Coordinate system used in the analysis of wavefront aberrations.

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axis of the optical system, A is the axial point of the exit pupil and 0, the paraxial image. 0 is the Gaussian image of an extra-axial object point and 0,O is the Gaussian image plane. The obliquity of the principal ray A 0 is measured by the intercept O,O, which is expressed as a fraction cr of the maximum value of 0,O. P is a point in the pupil and is defined in polar coordinates r (its perpendicular distance from A 0 expressed as a fraction of the maximum semi-aperture) and 4 (the angle between the planes P A 0 and A0,O). The aberration function W(a, I , 4) (the departure of the wavefront from a reference sphere) of an optical system having rotational symmetry can be represented by the sum of a series of terms, each of which is the product of a coefficient and powers of r2, a2 and 01 cos 4. If the reference sphere considered has its center at the Gaussian image point, the aberration function has the form

W ( o ,I , 4) = 0 ~ p 0 ~ 4 + 0 ~ 3 1cos ~ ~4 3 + 2 ~ 2 2COS’ ~ 24~ 2 +2c20cr2Y2+3cllfJ3Y cos 4

+ . . ..

(2.6)

As is well known, the general aberration terms can be classified into three categories: spherical aberration independent of 4 comatic aberration dependent of odd powers of cos 4 astigmatic aberration dependent of even powers of C O S ~ . Thus the ‘symmetrical’ part of the aberration involves spherical and astigmatic aberration, while the ‘asymmetrical’ part comprises comatic aberration. We can now represent the total wavefront aberration by

W(‘>

$1

= Weven(a,

I , +)+Wodd(a,

4))

(2.7)

where terms involving even and odd power of cos 4 respectively have been summed. 2.2.2.1. L a t e r a l s h e a r i n t e r f e r o m e t r y

As any point on a wavefront can be compared with a point on any zone of an identical wavefront by altering the shear between them, the shear interferometric principle makes possible a great simplification of the process of analysis of the interferograms. BATES[1847] showed that separate determinations of the comatic and astigmatic parts of the total wavefront aberration W ( o ,Y, +) from analysis of shearing interferograms are possible, and derived expressions for thcse cases. BROWN [1954] has also treated some examples of the calculation

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of the primary aberrations and showed that in cases where astigmatism and coma are present with their axes coincident - as for the case of lenses tested off axis - a pair of interferograms taken at the same focus can yield simply, by measurement of the change of tilt and curvature of the central fringe, a measure of astigmatism and coma. Essentially, the interferometer described by BATES[1947] consists of two plane parallel dividing plates S, and S,, and two plane mirrors (see Fig. 2.3). A convergent wavefront W from the system being tested is divided, and two identical, coherent wavefronts emerge

Fig. 2.3. Mach-Zehnder type shearing interferometer.

Fig. 2.4. SimplifiedMach-Zehnder type shearing interferometer.

after S,. An eye placed t o receive these emergent wavefronts will see two apertures laterally sheared in respect to one another. The magnitude of the shear is continuously variable by rotation of shear plate S, about a suitable axis perpendicular to the plane of the figure. The instrument can be regarded as a form of Mach-Zehnder interferometer. Two further plane parallel plates are added to preserve white light compensation; one, C,, rotates with S,, and the other, C,, rotates at twice the rate in the opposite direction. Bates pointed out that a simple variation in the optical system will rotate one wavefront

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about a principal ray, and that gives further information about the asphericity of the wavefront. Thus a rotation by 7212 will double the errors of astigmatism, and a rotation by n will double those due to coma. DREW[1951] introduced a simplified construction (see Fig. 2.4), consisting of a plane parallel dividing plate and two plane mirrors, M, and M,. Rotation of M, through an angle 8 results in shear through an angle 20 between displaced sights of the same wavefront. Orientation of mirrors MI and M, parallel to and equidistant from the plate ensures equality of path length in the two circuits, and rotation of mirror M, about a vertical axis through the point of incidence of the principal ray results in rotation of the principal ray about this point, but without change of path length. Thus when the apex of the cone associated with the principal ray is located on M,, this mirror may be rotated whilst identical path lengths are retained for individual rays.

Fig. 2 . 5 . Cyclic shearing interferometer.

BROWN[1954] made a wavefront shearing interferometer in the form of a cemented optical unit, which introduces a fixed shear and tilt between the interfering wavefronts. This construction results in an instrument which maintains correct compensation and may be very easily and quickly set UP. The instrument is optically a Drew type interferometer. A modification of this instrument has been introduced by DONATHand CARLOUGH[1963]. They obtained a variable shear with a pivotable mirror (corresponding t o M, in Fig. 2 . 4 ) , separated from the cemented prism unit by an oil film. The resemblance of the above mentioned instruments to the MachZehnder apparatus is striking. BROWN[1955] has designed some shearing interferometers related to another category, viz. the Jamin

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type interferometer. One design is very similar to that shown in Fig. 2.7, but with a small angle prism replacing the meniscus lens. However, by inserting a prism a shear is introduced which varies across the field of view. Another variant, in which the necessity of equalizing the two optical paths has been eliminated, was introduced by HARIHARAN and SEN[1960a]. The optical system of their instrument consists of a simple cyclic shearing system: a plane parallel plate S, with a semireflecting coating on one face, and three plane mirrors (see Fig. 2 . 5 ) . A convergent wavefront from the system under test is divided by S and the wavefronts are brought to a focus a t the same point on M,. If M, now is rotated by an angle 8 about a vertical axis passing through the focus point, the two cones emerging from the interferometer will be rotated in opposite directions by an angle 28 about this point (represented by dashed lines in Fig. 2 . 5 ) . An eye a t 0 will then see two displaced images of the system under test, sheared through an angle 40. I n order to obtain fringes of high visibility by the interferometers mentioned in this section a restricted source of light is necessary; the source may be extended provided the slit image direction is parallel to the axis of rotation of the shear plate. 2.2.2.2. W a v e f r o n t r e v e r s i n g i n t e r f e r o m e t r y The problem of obtaining the individual wave aberrations can be considerably simplified, if the number of aberration coefficients in eq. (2.6) to be evaluated a t one time is reduced by isolating either the symmetrical or the asymmetrical part of the total wave aberration. We will now see how the wavefront under test can be compared with its own mirror image directly, and how the term in eq. ( 2 . 7 ) can be measured separately. GATES[1955] and SAUNDERS [1955] introduced interferometers so arranged that one part of the wavefront from a small light-source is reversed, and then both parts are passed through the system to be tested along similar paths. A second reversal of one of the two parts of the original wavefront brings the two into coincidence so that interference can occur, but because of the reversal, onIy the asymmetrical variations of path in the optical system are detected. GATES[1955] used a Kosters double-image prism in an arrangement shown in Fig. 2.6.a. Successive positions of the original wavefront X Y from a small source a t S, are indicated in the figure. An eye

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looking into the interferometer prism from S, sees two images of the pupil of the system under test, which are mirror images and which may be made to overlap by rotating the prism about an axis perpendicular to the plane of the figure. Rotation of the mirror M about an axis with the same orientation produces opposite tilts of the two reflected wavefronts, which provides a control of the separation of the interference fringes. Gates showed how one interferogram is sufficient to determine the total comatic contribution a t each point in the pupil, and how the wave aberration can be broken down into its constituent terms. However, this method does not permit off-axis testing of lenses with one conjugate a t infinity (SAUNDERS [1957J).

M

__-- _-

-

!%gj 8:i

- _

-

---___

.

M

Qz

(b)

Fig. 2 ti Kosters type wavefront rcversing interferometers: (a) Kosters doubleimage prism interferometer; (b) Modified Kosters prism interferometer.

SRIJNDEKS [1055] has used a wavefront reversing interferometer with a modification of the Kiisters double-image prism. The apparatus, which is designed especially for testing lenses and concave mirrors, is shown in Fig. 2.6.b. Here the prism base is curved, which causes the part of thc wavefront coming from the object below the dividing plane of the double-image prism to appear folded onto the part above, after the second passage through the prism. If the dividing plane does not cut through the center of thc mirror or lens being tested, the folded images arc u n ~ q u a land fringes appear only in the overlapping area. Saunclers (MOLLET [lOGO]) has described many applications of

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this interferometer, e.g. to test an astronomical objective, reflector or refractor; one needs only to direct the telescope toward a star, with the prism replacing the eyepiece (the wavering of the interference fringes during such a test is a measure of the seeing condition, or inhomogeneity of the atmosphere through which the light travels). A unique method of analyzing the data from the interferograms is [1962]. The analysis, similar to that described given by SAUNDERS in sec. 2.2.1, involves a set of linear simultaneous equations. 2.2.2.3. R a d i a l s h e a r i n t e r f e r o m e t r y

A laterally sheared wavefront interferogram has some limitations: no sensitivity to wavefront slopes perpendicular to the shear direction, the interferogram covers only a part of the aperture, no sensitivity to periodic errors having a period equal to the shear in the direction parallel to it. These difficulties can be overcome by taking a number of interferograms, but they are eliminated with a new type of shearing interferometer. I n this, instead of obtaining interference between two equal-sized images, one of the images is expanded. When the centers of the two images coincide, shear occurs in the radial direction. Under these conditions, the interferogram always covers the entire area of the wavefront under test. An interferogram obtained with this type of interferometer can be interpreted with comparative ease. In this case, if W ( r ,r$), the departure of the wavefront a t a point ( Y , 4) in the exit pupil from the reference sphere (with its center a t the focus of the converging beam from the system under test, and passing through the axial point of the pupil) is given by eq. (2.6), then the path difference at the corresponding point in the interferogram is given by

A W ( ~ ’r$),

=

w(Y’,+w(sY’,

+3c11(1-s)Y’

r$)

= 0c40(1-s4)~’4

cos r$+ 2 ~ 2 (1 2 -s’)Y’~ cos r$+ . . .)

+ 0 ~ 3 1 (1- s ~ ) Y ’ ~

COS’

4+ 2cZ0 (1-S’)Y’’

(2.8)

where the shear factor s, is the ratio of the diameter of the inner cone to that of the outer cone emerging from the interferometer (cf. HARIHARAN and SEN [1961]). Eq. (2.8) shows that each of the individual aberrations of the wavefront under test is reproduced in the interferogram unchanged, except for a scale factor. To evaluate the wave aberration coefficients, the interferogram is analyzed according to the usual procedure. This gives the values of the terms 0c40(l-s4),

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6cal(l--s3), . . . etc., from which the values of the coefficients o ~ 4 0 , o ~ g l ,. . . can be readily calculated. Quantitative measurement of the aberration coefficients of objectives, especially microscope objectives, have been discussed and performed with a radial shear interferometer by HARIHARAN and SEN

Fig. 2.7. Jamin type radial shearing interferometer.

& & e i l , (b)

Fig. 2.8. Kosters type ‘exploded’ shearing interferometer: (a) Outgoing light path; (b) Return light path.

[1961, 19621. Measurements can be made off axis as well as on axis. The variation of all the aberration terms with wavelength can also be examined with this instrument. Since testing is here carried out under conditions approximating those under which the microscope is

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really used, the effect of changes in various parameters can be studied quite readily. BROWN[l959] introduced an interferometer consisting of a pair of plane parallel plates of equal thickness in a Jamin type arrangement between which are placed a lens and a plane parallel compensator (Fig. 2.7). The lens acts as a telescope of near unit magnification. The instrument is placed at the focus of the system under test. Another type, with ‘exploded shear’ (s m 0), consisting of a Kosters prism instrument with the addition of a pair of lenses (see Fig. 2.8) has been described by BROWN[1962]. As the expansion here is very large, the wavefront aberrations appear unmodified in the interference pattern. The same effect has been achieved by DYSON[1957], who, instead of a Kosters prism, used a double focus lens in the form of a

Q

Fig. 2.9. Triangular path radial shearing interferometer.

symmetrical triplet, consisting of a central double concave lens of calcite and two double convex glass lenses, as beam splitter. This triplet, which divides light polarized at 45’ with the.optica1 axis of the birefringent lens (lying in the plane of the lens) into two beams, was so designed as t o have zero power for the ordinary rays and a focal length of a few centimeters for the extraordinary rays. If the system being tested is a concave mirror with its center of curvature in the center of the triplet, the ordinary rays are imaged by the mirror t o a diametrically opposite point in the triplet. A A/4 plate, placed between the triplet and the mirror, with its principal axes at 45’ to the optical axis of the lens, interchanges the polarization directions, so that the returning ordinary rays become extraordinary

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and are refracted to a focus before the lens. The extraordinary beam, on the other hand, is refracted at its first passage through the triplet, fills the whole aperture and is then reflected to a focus, which, because of the symmetry of the system, coincides with that to which the initially ordinary rays were focused. I n this way the two emergent beams will coincide in the absence of errors in the concave mirror, and are able to interfere by means of an analyzer. This interferometer was used for testing objectives and mirrors. In this type of instrument it is essential to use a double pass system in order to maintain equality of brightness in the two interfering wavefronts. Transverse movement of the beam splitter introduces tilt into the interference pattern, whilst circular fringes are produced by axial displacement. HARIHARAN and SEN[1061] have used another kind of instrument, the optical system of which is shown in Fig. 2.9. In this shearing interferometer the converging beam from the system under test is brought to a focus at a point on the beam-dividing plate, S. Two lenses, L, and L,, are introduced into the interferometer path in such a way that their principal foci lie on the plate S. Under these conditions, the two beams diverging from a point on S are brought back to a focus once again at S, before emerging from the interferometer. If the focal lengths of the lenses differ, the two cones will be of different angles. An eye placed a t 0 will see two superposed concentric images of the system under test, the inner one of which will, in general, be crossed by interference fringes. The permissible light source for this kind of shearing interferometers is a small pinhole. 2.2.3. Measurement of optical transfer characteristics

The treatment of image formation from the standpoint of Fourier analysis ha5 led to the characterizing of optical systems by their response to contrast as a function of spatial frequency, called transfer factors or modulation transfer functions, which is now probably the most useful single test of optical system performance, since it relates the system to both design and image-forming properties under any chosen conditions. The transfer function is simply the inverse Fourier transform of the intensity distribution in the image of a point source. As there is no loss of generality in considering the more convenient case of unidimensional structures, they are mostly used in this kind of measurement. The normalized transfer function in the plane of shear is generally complex and may conveniently be written in the form

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D ( s ) = T ( s )exp (iO(s)>,

53

(2.9)

where T ( s )is the transfer factor modulus (modulation transfer factor) and O(s) represents a spatial phase shift. The basic feature of these methods is that a collimated light beam from the lens under test is directed into the interferometer, where a beam splitter divides it into two beams, which are laterally sheared with respect to one another; interference taking place in the region of overlap. If the pupils are circular and taken to be of unit radius, the relative shear, s, is numerically equal t o a normalized spatial frequency parameter. The total flux in the interference pattern is given by (2.10) @(p) = @,{l+I',,T(s) cos [(?-O(S)l}, if we make the simplifying assumption that the amplitudes of the two beams are equal and uniform over the whole area of the two pupils. di, is the total flux through the pupils, 1'21the coherence factor for the two beams, and q the phase difference introduced between them. Eq. (2.10) shows that, as q is varied linearly with time, the total light flux in the interference pattern consists of a constant and a cosinusoidally varying part, whose amplitude and phase give, respectively, the modulus T ( s ) ,and argument O(s), of the transfer function for the frequency s. Thus, the modulus and argument of the transfer function of the optical system may be obtained directly from the amplitude and phase of the resultant harmonic variation of the total flux in the interference pattern. The use of a wavefront shearing interferometer for the measurement of the optical frequency response was first suggested by HOPKINS [1955]. BAKER[1955] constructed an interferometer of that kind and used it to verify theoretical curves for a defocused aberration-free system. KELSALL[1059] showed how such a n arrangement can be modified to make automatic recordings. Besides verifying the response curves for defocused systems he also verified a large number of curves relating to spherical aberration. HARIHARAN and SEN [1960b] also described measurements of defocused optical systems. TSUKUTA [ 1963al has made measurements on microscope objectives free from aberrations and also those with primary spherical aberration. He has also performed measurements of photographic objectives and compared them with those obtained by geometric optical calculation from the lateral aberrations which are experimentally determined from the deformation of interference fringes (TSURUTA [1963b]).

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A simple modification of the Michelson type of interferometer, in which the plane mirrors have been replaced by roof mirrors, in order to provide for a lateral shear of the wavefronts coming from each

Fig. 2.10. Michelson type shearing interferometer for measurement of the optical transfer function. 1-

Fig. 2.1 1. Triangular path shearing interferometer for measurement of the optical transfer function.

branch of the interferometer, was used by BAKER[1955]. Lateral displacements of the mirrors cause shearing. cy was varied by moving one of the roof mirrors in a direction parallel to the incoming beam.

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The readings were performed with a two-beam flicker photometer, in which one beam contains the lens under test and the other a good quality collimator. KELSALL[1959] used corner cube prisms in place of plane mirrors in a Michelson interferometer. The moving parts employed for wavefront shearing and phase changing were mounted independently of the interferometer bed (cf. Fig. 2.10). The relative shear, s, between the two wavefronts was produced by two plane parallel identical plates placed at right angles one in each arm of the interferometer and which could be rotated about a vertical axis. With this arrangement, a lateral shear is introduced without relative tilt between the wavefronts, leaving the phase difference unaffected. To produce the cosine signal, implied in eq. (2.10), a, was changed linearly with time by causing a wedge prism of small angle to reciprocate in one of the interferometer arms using an Archimedean cam. Automatic recording of the transfer function (as a function of spatial frequency) is achieved by oscillating the wedge prism and recording the a x . signal from a photomultiplier on a chart paper, the motion of which is linked synchronously to the rotation of the two plane parallel plates. MONTGOMERY[ 19641 has used a modified Drew type interferometer (cf. Fig. 2.4) to measure the modulation transfer function of a lens. HARIHARAN and SEN [1960b] have used a triangular path interferometer, shown in Fig. 2.11. A collimated beam from the lens under test, which is incident on the semi-reflecting mirror, S, is divided into two beams, which traverse a close triangular path in opposite directions before they are brought to a focus on a diffuser, placed in front of the photocathode of a photomultiplier. By this arrangement it is possible to shear the wavefronts without introducing a path difference between them, merely by moving the mirrors MI and M, as a unit. However, a more convenient and sensitive method is to use a tilted, plane parallel plate, introduced in the path of the two beams (cf. Fig. 2.11), which shears the beams by an equal amount in opposite directions (HARIHAKAN and SIKGH(19591). The phase difference p is introduced by a polarizing system, which permits a linear variation of the phase difference over exactly 237 by the rotation of a single polarizing element. The two wavefronts emerging from the interferometer are polarized in mutually perpendicular planes by means of the polarizers P, and P,, which have their planes of oscillation parallel and perpendicular, respectively, to the plane of the figure, and a half-wave plate having its slow axis at 45' to this plane, if the

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original wavefront is polarized at 45" by means of another polarizer, P, . The emergent wavefronts become circularly polarized by means of a quarter-wave plate with its slow axis a t 46", and the polarizer I?, , which follows, introduces the phase difference v, if its azimuth makes an angle p/2 with the planc of the figure. A shearing interferometer using polarized light, for the measurement [1963a]. A of the transfer function, was introduced by TSURUTA birefringent system, consisting of a plane convex and a plane concave lens cut parallel to the optical axis and mounted together with the axes crossed (the cemented planc siirface of a Wollaston prism is replaced by a spherical one), is employed for obtaining a variablc amount of shear of the aperture (cf. Fig. 2.12). This Wollaston lens is placed between crossed polarizcrs at the focus of the convergent beam from the objective under test, and a desired amount of shear between the ayerturcs is obtained by the displacement of tlie Wollaston lens normal to the incident beam. In order to obtain a greater amount of light (cf. 9 4),the system is symmetric about the object plane of the microscope objective (see Fig. 2.12). The identical Wollaston lenses are placed perpendicular to each other, so that the axis of the convex lens is parallel to that of the concave lens of the other. To measure transfer functions of suitable azimuth, the polarizer P, and the Wollaston lcnscs are rotated to the corresponding angle and a 212 plate is rotated to half an angle of their revolution, in order to leave the planes of polarization at the analyzer P,, unchanged. The phase difference, 9,is varied by a Soleil-Babinet compensator SB placed before, and a t an angle of 45" to the last polarizer.

Fig. 2.12. Compensated polarization interferometer for mcasurement uf tlie optical transfer function.

TSURUTA [ 1963bI has also introduced another polarizing interfero-

meter, in which a modification of the Savart polariscope (cf. $ 4) is employed. I n order to obtain a continuous variable lateral shift s of the aperture of the objective under test, an arrangement is used which permits a changeable thickness of the birefringent crystal plate. Four quartx wedges identical in shape cemented two and two with their

11,

§ 21

0PTI

cA L c 0 M P 0N E N Ts

57

principal sections at right angles form the modified Savart polariscope MS (cf. Fig. 2.13). The wavefront from the lens system under test is polarized in such a way that the oscillation plane bisect the two principal planes of the wedges and passes after collimation through the combination of the two birefringent prisms MS (the shear is changed by sliding the prisms in opposite directions) and an interference pattern, which is localized in the region of overlap of the laterally sheared images of the exit pupil, becomes visible by means of an analyzer P, (P, is used perpendicularly to PI). The phase difference p is varied by means of a Soleil-Babinet compensator SB placed between the beam splitter and the analyzer and with its axes parallel to those of the prism combination. The modulus and phase of the transfer function given in eq. (2.10) for the lens under test are given by changes in amplitude and phase, respectively, of the total flux in the interference pattern when variations of w are performed. MS

Fig. 2.13. Polarization interferometer for measurement of the optical transfer function using a modified Savart polariscope. 2.3. EXAhIINATION O F DIFFRACTION GRATINGS

A wavefront shearing interference test has been applied to the study of diffraction gratings (plane or concave) by GUILD [1957], who dealt with those properties of the gratings which affect image formation. The test reveals the errors of figure of the diffracted wavefront from a concave grating, or a plane grating with a collimating lens or mirror, so oriented that one of the spectral orders is returned to the observation point. The beam divider was a Wollaston prism of small divergence combined with a polarizer, cemented to the face of the prism nearest the light-source with its polarization plane inclined a t 45" to the plane of divergence in order to ensure similar polarization of the returned wavefronts. Both the outward and return beams pass through the beam divider in order to give a high field brightness (cf. 9 4 and Fig. 4.4). Thus a small mercury bulb is mounted just behind the beam splitter below the middle of its aperture, leaving the upper half clear for the returning beam to come through to an

58

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INTERF'BROMETRY

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2

observing telescope or camera. The relation, which in this case determines the path difference A W of the sheared wavefronts a t any point of the superposed field, is

AW = 2 v ( h - y d / r )

(2.11)

for paraxial rays, when the surface under test is of perfect figure. y is the distance from a plane perpendicular to the shear direction containing the axis of the optical system, h the distance from this plane of a level in the beam divider at which rays are divided without differential retardation of their divergent components, d the distance of the beam splitter from the focal plane, z, the angle of divergence of the beam divider and r the distance of the surface under test from the definite plane of the beam splitter (when a collimating lens is used - distance from the lens to the beam splitter). If the direction of the shear is parallel to the rulings and the rulings are straight and parallel, the path difference between corresponding points will depend only on the figure of the surface, and if this is perfect the fringes will be straight, parallel and equidistant. If, on the other hand, the shear is inclined t o the rulings, the separation of corresponding points will span, in general, some integral number of rulings plus a fraction: N + f , where f is a proper fraction. For the wavefront of the mth order a path difference of &m(N+f)1 is introduced in addition to the basic path difference given by eq. ( 2 . 1 1 ) :

AW

=

2v(h-yd/r)fm(N+f)1,

(2.12)

where 1 is the wavelength of the light. For a given shear, the ruling spanN+f varies inversely as the pitch of the grating, so any variation of pitch from one part to another will produce a corresponding displacement of the fringes from their ideal courses. Curvature of the rulings causes vertical variation of f , which is shown by non-uniform spacing to the fringes. Non-parallelism of the rulings, usually called fan error, changes the slope of the fringes and, if unevenly distributed along the grating, produces local or general curvature. When a grating is so oriented that one of the spectral orders is returned to the observation point, the test reveals the errors of figure of the diffracted wavefront. Usually local displacements of rulings from their ideal positions are expressed in fractions of the average ruling-width and local displacement of fringes from their ideal positions as fractions of the average fringe width. I n these units the

11, §

31

ILLUMINATION PROPERTIES

69

fringe displacements are equal to the ruling displacements multiplied by the factor 2m sin un for periodic type of error (K is the ratio of the component of the shear perpendicular to the rulings to the period of the error); m for unique local flaws; mz/2 (average) for irregular non-periodic errors, when the fringes are observed in the mth order. Thus, by rotation of the beam splitter, the effect of a periodic error relative to non-periodic errors or periodic errors of other frequencies may either be augmented or reduced.

3. Studies of Illumination Properties 3.1. MEASUREMENT O F COHERENCE

A wavefront shearing interferometer may also be used as an instrument for measuring coherence. If, in an'optical system that gives rise to an intensity, I(x),and a coherence, r ( x l , xz), a shear, s, is introduced before the x plane, then the intensity there becomes

I,@) = I(x)+I(x-s)+29e{I'(x,

x-s)}.

(3.1)

According to this relation the coherence a t any plane in an optical system may be measured as an intensity by inserting a shearing equipment in front of that plane. The direct intensity given by eq. (3.1) yields the real part of the coherence, while the intensity with a 4 2 path difference introduced between the two beams would yield the imaginary part (STEEL[1958]). 3.2. DETERMINATION O F SPECTRAL PROFILES

Interference fringes produced in an interferometer are physically determined when the properties of the interferometer and the spectral distribution of energy are known. Conversely, the study of fringes yields information about spectral distribution when properties of the interferometer are known. GIRARDet al. [1959] have used a shearing interferometer, similar to the polarizing interferometer presented in Fig. 4.4,for spectroscopic measurements. They measured the signal from the interferometer as a function of the path difference A W between the interfering beams. d W was varied by displacement of the beam splitter perpendicular to the light beam direction in the plane of shear. This method gives a record of the Fourier transform of the spectrum of the light source. By inverse transformation the spectral profile is then found.

60

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[II,

9

3

ERICSSON and SJOFALL [l960] used the same kind of interferometer, in which the interferences obtained between light reflected from different gauge blocks are recorded. From these the visibility V for some fixed interference orders is obtained. The spectral profile f ( k - k , ) is then calculated using f(k--KO)

+cc

=

J V ( d W ) cos {2ndW(k-k0)}dd W ,

-0a

(3.2)

for cases where f is symmetrical about the wave number K O . 3.3. STUDY O F SCINTILLATION

Danjon (BEER[1955]) has used a shearing interferometer like that in Fig. 2.3 for observation of atmospheric disturbances. The interferometer is placed at the eye-end of a refractor and the system of fringes in the area common to the two separated beams from the star light is examined. The fringes are disturbed by the inhomogeneity of the atmosphere, i.e., by the resultant differences in refractive index along the path of any two interfering rays. If the distance d , between these rays is not greater than several centimeters, the amplitude of the disturbance is proportional to d, but for values larger than a certain limit, the motion reaches a maximum and does not increase further (the fluctuations for elements wide apart are uncorrelated). The absolute value of the turbulence can be derived from measurements of the movement of the fringes for small values of d. The same instrument also allows measurements of the visibility of the fringes. 3 4. IRSAGE POSITION MEASUREMENT O F SMALL LIG IIT SOURCES

The problem of measurement of the position of a small image may be considered equivalent to the determination of a path difference d W. Several types of wavefront shearing interferometers can be used, but a simple Wollaston prism is convenient (DYSON[1963]). The departure of a small light source image from some datum position can be ascertained by shearing the wavefront, giving rise to the point image, in the exit pupil of an imaging system of angular aperture GC into two surfaces separated by rotation about an axial point. If the shear consists of a rotation about the axial image point, the two wavefront surfaces rcmain parallel to the first order of approximation, but separated in the axial direction. The path difference A W on the axis between the two identical wavefronts is given by:

dW = he,

(3.3)

LARGE

TRANSPARENT

OBJECTS

61

where h is the distance of the image point from the axis and 0 is the angle of shear. The view seen through the interferometer is a double image of the aperture and it is evident that 8 cannot exceed M , as the path difference measurement can only be effected in the area of overlap. As the path difference can be measured with great precision, it is clear that very precise determination of h is possible. A practical method of doing this has been described by DYSON[1963]. The method may be applied to positional astronomy and industrial metrology. 4. Examination of Large Transparent Objects For examination of extended transparent isotropic objects, shearing interferometers using polarized light have been found suitable, for they are easy to adjust, insensitive to vibration and work with a very high precision. The essential part of a polarization interferometer is a birefringent system, which doubles the incident wavefront, the shape of which is modified by the optical path due to the object, into two identical wavefronts. The two waves are rendered coherent by a polarizer, and the corresponding vibrations are rendered parallel by an analyzer, so that they are able to interfere and to transform the optical path difference variations in the object into variations of intensity. The birefringent systems employed are mainly of the Wollaston prism and Savart polariscope types. The Wollaston prism, which introduces an angular shear (see, for instance, BARTHOLOMEYCZYK [1960]), consists of two prisms of a uniaxial crystal, cemented to a plane parallel plate. The optical axes are parallel to the faces and crossed (see Fig. 4.l.a). The Savart plate, which gives a lateral shear (see, for instance, FRANCON [1952a]), is composed of two plates of equal thickness of a uniaxial crystal (quartz or calcite) cut at 45" from the axes and crossed (see Fig. 4.1.b). In Figs. 4.2.a and b the main arrangements using these elements are shown (see, e.g. FLUGGE [1956]). A collimator C, with an illuminated slit in its focal point illuminates the object A. The beam passes then through the Savart plate S, placed in parallel light (see Fig. 4.2.a). The objective 0 gives an image of A in A'. The polarizers P, and P, are situated anywhere, but so that S is between them. In the arrangement employing a Wollaston prism, W (see Fig. 4.2.b), this prism is placed at the focus of the objective, 0, and conjugate to the entrance slit.

62

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5

4

The Wollaston prism gives perfectly rectilinearly localized fringes, but not a uniform field. The Savart plate, on the other hand, gives a perfectly uniform field, but fringes a t infinity which are not perfectly

$

4

b 2

(c) (d) Fig. 4.1. (a) Wollaston prism; ( b ) Savart polariscope; ( c ) Compensator with uniform field; (d) Polariscope with rectilinear fringes.

(b) Fig. 4.2. Polarization interferometers: (a) 'Interferometer with a Savart polariscope; (b) Interferometer with a Wollaston prism.

rectilinear. In general, this is not a disturbing factor. However, when working with a very wide field of view, these inconveniences can be avoided by using a modified compensator.

11,

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An achromatic compensator of the Wollaston type with a considerable field is obtained if a 4 2 plate and a plane parallel plate, of the same birefringent material as the prism and with a thickness exactly equal to half the total thickness of the first prism, are placed at each side of the Wollaston prism (see Fig. 4.l.c in which the orientations of the optical axes are indicated) (FRANFON and SERGENT[1955]). Likewise, the Savart fringes can be made rectilinear by turning one of the subplates 90" in respect to the other and interposing a 4 2 plate, whose axis is at 45" to the axes of the plates (see Fig. 4.1.d) (FRANCON [1957]).

A

Fig. 4.3. Polarization interferometer working by reflexion and using a Savart plate.

In cases of great magnification, and in high-speed photography it might be valuable to avoid the use of a narrow entrance slit and hence low field brightness. There are several ways of achieving this result. One is to place a second Wollaston prism in the slit plane instead of the entrance slit in Figs. 4.2.a and b. Another solution is to place a second Savart plate between C and A (see, e.g., FRANCON [1957]). The second beam splitter is oriented in such a way with respect to the first one as to have the birefringences subtract one from another. In either case,

64

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INTERFEROMETRY

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3 4

P, must be placed in such a way that the two birefringent systems are between PI and P,. In practical designs of these interferometers the object is placed against a mirror and the object is twice passed through. Pig. 4.3 shows such an apparatus using a Savart polariscope (RENET [ 19561and F R A N ~[O 19571). N An opal lamp sends polarized light through

Fig. 4.4. Polarization interferometer working by reflexion and using a Wollaston prism.

Fig. 4.5. Polarization interferometer with two Wollaston prisms.

the polarizer P,, and it is reflected by n semireflecting plate. The light beam crosses the Savart plate and object A, is reflected back through the birefringent plate and then a lens, a polarizer, P,, and is finally coiidcnsed on the pupil of the observer. When using a Wollaston prism (Fig. 4.4),the transparent object, A, is placed near a spherical mirror (LENOUVEL and LENOUVEL [1938]). An image of the light-source is formed in the beam splitter, which is

11,

s 41

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TRANSPARENT

OBJECTS

65

placed near the center of the mirror, so that the reflected light in turn gives an image of the source in the beam divider. An objective 0 gives in A' an image of the object A. The double passage through the birefringent plate avoids the use of a slit, because it is equivalent to a setup with two beam splitters. In Fig. 4.5 two mirrors are used, so that the object is passed only once. These interferometric methods can be used for the analysis of gas flows in wind tunnels. They allow optical visualization to transonic and supersonic speeds, where it is usually necessary to extend the field of study to fast evolving phenomena by photography and high speed cinematography, and a t the same time the possibility of a quantitative interpretation. CHEVALERIASet al. [1957] used two

Fig. 4.6. Triangular path wavefront shearing interferometer.

optical setups, one (Fig. 4.5) for the over-all study of the flow around an airfoil at transonic speeds, the other (Fig. 4.2.b with a Wollaston prism instead of an entrance slit) for the detailed study of the flow between a shock wave and an airfoil at supersonic speeds. PHILBEIIT [ 19581 has performed quantitative measurements on different profile types with the arrangement in Fig. 4.5. Another type of interferometer has been described by HARIHARAN and SEN [1959]. They used a triangular path shearing interferometer consisting of a beam dividing mirror, made up of two identical plane parallel plates clamped together and with one of the inner faces half reflective, and two identical concave end mirrors (see Fig. 4 . 6 ) . Since a converging beam is used, the beam splitter may be quite small, while the concave mirrors can be of any desired aperture. An image

66

APPLICATIONS

OF SHEARING

INTERFEROMETRY

[11,

$ 5

of the entrance pinhole is formed on the beam splitter. This image is at the focus of the two end mirrors, so that the transmitted and reflected rays diverging from the beam divider traverse a closed triangular path in opposite directions before they are brought to a focus once again on the beam divider. By adjusting the beam splitter and the two concave mirrors, the two sets of rays formed at the beam divider can be made to traverse exactly the same path, or a shear may be introduced so that the two beams moving in opposite directions are laterally displaced with respect to one another in the space between the two end mirrors. Hariharan and Sen have found this interferometer particularly suited to such problems as the examination of large plates of glass and the analysis of flow patterns in wind tunnels. Two methods of observation are possible with these types of interferometers. In the first - the method of total doubling, particularly suited to the case of isolated details - the lateral displacement of the two images of the field under test is made large enough for each of the two images of the detail to interfere with the uniform background of the other. The fringes observed will then correspond t o lines of equal optical path. In the second - the differential method, convenient to use with extended or complicated objects - the lateral displacement is kept small compared to the width of the details which are being studied. The interference order at any point is then linearly proportional to the gradient of the optical path at this point in the direction of the displacement.

5. Studies of Physico-Chemical Phenomena in Liquids 5.1. MEASUREMENTS O F COEFFICIENTS D E S C R I B I N G TRANSPORT

PROCESSES

Shearing interferometry has been applied for the registration of the concentration distribution of levelled solutions and utilized to increase the precision of measurement of particular properties of liquids. The optical arrangement, which has been used for this kind of measurements, is based on the shearing interferometer shown in Fig. 4.2.a. A coherent, monochromatic wavefront is split up by a Savart plate after passing through a liquid gradient, and the interfercncc fringe system dependent. on the gradient of the refractive index appears within a sharp image of the object. In Fig. 5.1 a suitable setup is shown (INGELSTAM [1955, 19571, BRYNGDAHL 1119571). An optical rcduction R , is introduced, by using an afocal system made of

11,

§ 51

P H Y S I C O - C H E M I C A L P H E N O M E N A IN L I Q U I D S

67

two well corrected collimator lenses whose foci coincide, in order t o keep down the dimensions of the Savart plate. The shear s is parallel to the direction of the gradient (x direction) and perpendicular to the

Fig. 5.1. Wavefront shearing interferometer used for studying refractive index gradients.

entrance slit. For crossed polarizers the wavefronts interfere destructively in the image plane, if

AWjrlx

= mAR/s,

( m = 0, 1, 2 , . . .).

(5.1)

AW is the path difference between the two wavefronts sheared a small distance, A x = s, in the x direction (for convenience R = 1 is introduced in the following) and 1is the wavelength of the light. In some cases, where the mathematical form of the gradient is given by a solution of the partial differential equation

avjat = a ~ v ,

(5.2)

in which A is the Laplace operator, precision measurements of the coefficient a have been performed. 5.1.1. Determination of the diffusion coefficient

To produce a time-dependent interference fringe pattern from which the diffusion coefficient D can be deduced, a diffusion boundary, which is initially sharp, is formed between a solution and a solvent in a column with rectangular cross section. The change in concentration in the vicinity of the original boundary is followed by studying the changes in the refractive index gradient of the liquid column as a function of time and distance. For this case, I.' in eq. (5.2) is the concentration c of the solution in the distance x from the original boundary, and t is the time elapsed since the boundary was formed. The diffusion equation for one-dimensional free diffusion is

aqat = alax-pacjax).

(5.3)

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[11,

§ 5

In cases where the diffusion coefficient, D, is independent of the concentration, q. (5.3) may be written in the form

awjat

=

Da2wIax2,

(6.4)

if we introdiicc the optical path W ( x ,t ) = n(x,2) d rn (K,+k,c(x, t ) ) d , where 92 is the refractive index, d the thickness of the cell, and K , and k , are constants. The solution of eq. (5.4) for the given initial and boundary conditions is: (6.5)

if A W , is the difference in optical path between the two original solutions. A simple evaluation method (BKYNGDAHL [1962a]) is to introduce eq. (5.5) and the relation between A W / i l x and aW/ax in eq. (5.1), i.e. dW/Ax = const. The following expression is then obtained in the first order approximation:

where 7 = ( 2 ~ ) ~ z/ t=, ln(t/to),2x the distance between symmetrically situated fringes in the interference pattern, to an assumed time unit, At is n zero time correction and k is a constant. The diffusion coefficient U is then found from the slope of the straight line arising, when z is plotted as a function 7 ( A t is calculated from eq. ( 5 . 6 ) ) . This shearing interfcrometric method can also be applied to cases with a concentration-dependent diffusion coefficient, which produces a skewness in the gradient curve. As is 5een above, only the x value (position of the fringes) for a fixed value of 8Wjax is obtained from the interferograms. However, this does not limit the application of [1962]); eq. (5.3) can be the method (URYNGDAHLand LJUNGGREN reduced to an equation in one variablc only, by the substitution 5 = x/(av't). Then which gives

-2(dW/d[

djd[{L)(i)dW/d[},

(5.7)

(5.8)

For al.l//i)n. = const = K we obtain dW/d[

=

2kl/t,

(5.9)

11.

3

51

PHYSICO-CHEMICAL

PHENOMENA

IN

LIQUIDS

69

which means that if 2 k d t is plotted versus i', then dW/dt is given as a function of ( and we have a record over the whole gradient curve from the locations of the interference fringes. The diffusion coefficient D can now be determined from eq. (5.8). 5.1.2. Determination of the thermal diffusivity of liquids

The experiment is so arranged that from a given time an electric current is passed through a thin straight vertical wire, placed in a cell with a square cross section containing a homogeneous liquid, the thermal diffusivity of which is to be measured (BRYNGDAHL [ 1 9 6 2 b ] ) . Constant heat in the wire gives rise to a cylindrical temperature field in the liquid surrounding the wire. For this case eq. (5.2) may be written in cylindrical coordinates

z l a t = i/r - aja+aaT/at),

(5.10)

where T is the temperature rise in the distance Y from the cylinder axis after the time t in a liquid of thermal diffusivity a. If the heat production per unit of length is q and the thermal conductivity A, then the temperature gradient field is given by (5.11)

which corresponds to a refractive index gradient (5.12)

as the refractive index n is a function of the temperature T and pressure 9, and f~ in this case is constant. According to the two-dimensional refractive index distribution, the optical path gradient is obtained from an integration over the cell thickness d : (5.13)

Introduction of r2 = x 2 + z 2 , &/ax = X / Y and eq. (5.12) into eq. (5.13) yields aW X exp {- (x2+z2)/4at}dz, (5.14)

ax

if an/aT is regarded as constant. When d is sufficiently large, so that

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SHEAKINO INTERFEROMETRY

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5

5

the walls of the cell have no influence on the temperature gradient, the solution of eq. (5.14)has the following form:

and

aw = - (”) 4 - {-1-Erf --

ax

,aT D 2 A

{5.15) (x/l/4at)}

for x

< 0.

Thermal diffusivity can be determined from the changes of the interferograms with time. If the distance 2xi between symmetrically situated fringes is plotted versus t , each fringe pair corresponds to a straight line. The slopes of these lines (zero time correction is here uninteresting) are used for the calculation of a. According to eq. (5.1), d W / d x is increased with a known fixed value, when going to the next fringe pair, and from eq. (5.15) it is clear that x/1/4at will be a constant for each fringe pair. If we set di, = Erf ( x i / l / 4 a t ) -Erf ( x J d 4 a t ) and if x i / l / t is known from the experiments, 8, can be considered as a function of a. If only three fringe pairs, i.e., xl,x2 and x 3 , are used, then

&(a)

= &(a),

(5.16)

which is a transcendental equation for the determination of a. The advantages of this method are that convection flows are easy to discover and that the registration of the temperature field is performed far away from the wire and no apparatus constants or dimensions enter into the calculations. 5 2 . RECORDING O F THE KEFKACTIVE INDEX VARIATIONS I?$

LIQUIDS

The use of shear interferometers is particularly recommended, where the form of the refractive index gradient curve is the kind of information that is wanted, which is often the case in diffusion, electrophoresis, ultracentrifugation, sedimentation, heat conduction, heat flow, and measurement of optical inhomogeneity. The wavefront shearing interferometer illustrated in Fig. 5.1 has been enlarged to a direct derivative recording method (BRYNGDAHL and LJUNGGREN [1960], BRYNGDAHL [1963]) by adding a second beam splitter in convergent light, with its shear in a perpendicular direction in respect to the first shear. I n Fig. 5.2 is shown a n arrangement

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containing two beam splitters of the kind shown in Fig. 4.1.d (other alternatives are described by BRYNGDAHL C19631). The fringe system appears within a sharp image of the object under test and the displacement of each fringe represents the derivative of the optical path in Cartesian coordinates for that part of the object through which the fringe passes.

Fig. 5 . 2 . Refractive index gradient recording wavefront shearing interferometer.

In a method described by WEINSTEIN[1953], records of both concentration and concentration gradient are obtained simultaneously as interference fringes superimposed on an image of the cell containing the liquid. He used an arrangement similar to that in Fig. 4.4. The spherical mirror is replaced by a plane mirror and a collimator and a double cell is placed between them containing liquid with varying concentration in one compartment and homogeneous liquid in the other. If the doubling angle arising from the Wollaston prism in the horizontal plane is suitable, two images of the cell will partly overlap. In the region where the images of the two compartments overlap, the fringes map the concentration in the cell. If, in addition, the Wollaston prism and the polarizer with its principal direction at 45" to those of the beam splitter are turned through a small angle about the optical axis, the lateral shear will become a small vertical component; then, in the region where the images from the same compartment of the cell overlap, the fringes map the concentration gradient. The horizontal scale of the integral and gradient patterns can be varied simultaneously by moving the Wollaston prism along the axis, while the scale of the gradient pattern alone can be varied by rotating the prism. 6. Examination of Microscopic Objects 6.1. APPLICATIONS I N MICROSCOPY

The wavefront shearing interferometer principle can be used advantageously in conjunction with the microscope, enabling measure-

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6

ments of small objects (transparent or opaque) to be made very rapidly and accurately. Phase changing specimens would give trouble with ordinary microscope, as they tend to disappear under full condenser illumination, but if viewed by a shearing interference microscope they present no particular difficulty; for this reason such interferometers have been widely used in microscopy. The special properties of the shearing interference methods can be summarized in the following way: (1) Phase changes introduced by an object can be made visible as differences of intensity or colour in the image using monochromatic and white light, respectively; (2) Accurate measurcrnents of path differences are possible. A wide variety of designs of shearing interference microscopes has been developed during recent years. A fundamental feature of all these microscopes is that the observer sees two superimposed fields of view. According to the way in which these fields differ, wc can divide such microscopes into two classes characterized by: ( 1 ) Lateral separation between the two images of the preparation. As regards the amount of shearing we distinguish between the method of total doubling and the differential method (cf. the end of 3 4); ( 2 ) Longitudinal doubling, i.e., in the direction of the optical axis of the microscope, whcre one of the images is out of focus. The applications of shearing interference microscopes are very numerous. Among the different research fields in which they have been successively used are: Biology (one of the important applications has been the study of living cells, where the interferomctric method allows the determination of concentrations of solid material within a cell and also of the total quantity present, both being possible while the cell is alive), Metallography, Crystallography (studies of growth of crystals), Chemistry, Physics on thin films (thickness and absorption of a thin layer can be measured simultaneously, see, for example VAN HEEL and WALTHEK [1958]; it is also possible to carry out measurements while depositing the layer), Physics on liquid surfaces (measurement of deformation of liquid surfaces and studies of surface tension phenomena, see, for example FRANCON and GANDON[1958]). Shearing microscopes have also been used in other research fields.

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6.2. LATERAL SHEARING INTERFEROMETERS

6.2.1. Wavejront shearing interferometry Interferometers resembling the Mach-Zehnder (BARRELL[19491 and DYSON[1960]) and Michelson (NOMARSKI and WEILL [1955]) type of instruments, as well as the one introduced by GARTNER[1958] consisting of a wedge prism (a fixed wedge or a variable wedge prism, which was formed between two glass prisms facing each other) in the back focal plane of the objective, can be used in a normal microscope. In these types of beam splitters it is necessary to introduce a slit perpendicular to the shear, and thus not the whole of the aperture of the illuminating system can be used. This limitation, however, may

t

Fig. 6.1. Jamin type interference microscope.

be avoided, as will be shown later, by the application of compensated polarization interferometers (suppression of diaphragm), which have proved to be most useful in interference microscopy. LEBEDEFF[1930] used the Jamin interferometer in a microscope. In order to use white light, two identical birefringent uniaxial crystal plates, cut at 45" from the axis and oriented in the same way, were inserted under the microscope objective, 0, so that the specimen, A, is placed between the plates (see Fig. 6.1). Light entering the first birefringent plate is split up into an ordinary and an extraordinary beam. A 212 plate introduced a t 45" in respect to the principal planes of the birefringent plates then interchanges the vibrations, so that the extraordinary ray becomes ordinary in the second plate and the ordinary becomes extraordinary. In this way there is a compensation of

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path differences and white light observations (interference between the two beams) are made possible, when this arrangement is placed between two polarizers, P, and P,. A mere inclination of the last birefringent plate is sufficient to vary at will the path difference between the two beams. The disadvantages of this system are not only limited to the interposition of a crystal plate between the preparation and the microscope objective, but are also found in the chromatism of the Aj2 plate. NEUGEBAUER[1955] oriented the two crystal plates at 180" from one another and removed the 4 2 plate. In such a way without interchange of the directions of vibrations an optical path difference is introduced, but this can be compensated by inserting one more birefringent plate in the light path before the other plates. This arrangement can also be constructed as an interference eyepiece; the compensation plate, of course, is here still inserted before the preparation. FKANCON [1952a] has introduced another beam splitter in this field. He used a Savart polariscope (cf. Fig. 4.l.b), which is obtained by turning one of the two birefringent plates described above go", so that their axes are crossed, and removing the A12 plate. When using this double plate, which automatically introduces a compensation of the path differences, the object can be placed in front of the first polarizer, that is to say, that it can be illuminated in natural light. The whole interferometer may be inserted into the eyepiece (cf. the upper part of Fig. 6.2), which can be adapted t o any kind of microscope. This eyepiece makes possible quantitative measurements if a compensation device is added to it, consisting of a rotatable glass wedge of an angle of only a few minutes of arc formed as a circular plate, with a circular hole drilled in its center (JOHANSSON and AFZELIUS[1956], INGELSTAM [1957]). When using white light and total doubling, the measurements are performed in the following way: Rotating the wedge changes the incident angle at the Savart plate and hence it is possible to compensate for a change in the interference pattern produced by the object. The optical thickness of the object is found from the different settings of the wedge, by which the two images of the object are made equal in colour to the reference area in the image, corresponding to the common part of the two images of the hole in the wedge. A disadvantage of this arrangement is the reduction of brightness due to the necessity of an entrance slit. FRANCON [1957] has introduced a compensation system, a second birefringent system before

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the object, in order to be able to use the whole aperture of the illuminating system and at the same time keep the advantage of an eyepiece, which does not modify the microscope. A solution is shown in Fig. 6.2, where a second Savart plate, S,, is inserted between the condenser, C, and the preparation, A (cf. 9 4,where other procedures also are mentioned). The two polariscopes, S,and S,,must be oriented at 180" one from another, and the thicknesses of the double plates

h

Fig. 6.2. Compensated interference microscope using two Savart plates.

must be such as to produce fringes at infinity, which have the same spacing in the pupil of the observer. For measurements of optical path differences a third Savart polariscope may be inserted after the eyepiece in the arrangement r19601). This shown in Fig. 6.2 (see, e.g., CATALANand FRANGON polariscope (P, is here mounted after this plate), placed in convergent light, superimposes the fringes at infinity on the images of the specimen. These fringes represent points of equal path difference between the

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two sheared wavefronts and their displacements permit the measurement of the optical thickness of the object. In the FranGon apparatus the rays in the image plane are laterally displaced. However, a lateral shear in the object plane is equivalent to an angular shear in the focal plane of the objective (LOHMANN [1954]), and thus the Savart plate in the image plane can be replaced by a Wollaston prism in the upper focal plane of the microscope objective.

;T;---'

d

1;' h \

!

Fig. 6 . 3 . Intcrfcrence microscopes using two Wollaston prisms: (a) Compensation of the birefriiigent elements performed in the pupil planes; (b) Compensation 01 the birefringent elements performcd in the object plane.

Also in this case it is possible to use a compensation arrangement, so that the use of a diaphragm is made unnecessary. SMITH[1960] and NOMARSKI[l952] used a second Wollaston prism in the lower focus of the condenscr, which is shown in Fig. 6.3.a. However, this arrangcment generally requires a special objective, as the Wollaston prism is placed in its focal plane. FRANCON and YAMAMOTO [1962] have described another system (see Fig. 6.3.b),which has the advantage that the positions of the birefringent elements are not fixed in reIatioii

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to the objective, 0, and the condenser, C. A Wollaston prism, W,, placed above the objective is imaged on a plane M, below the plane of the specimen A. Another prism, W,, placed below the condenser, is imaged above the focal plane of C. By adjusting the distance between C and 0 insuch a way that the image of W, coincide with M, a complete uniform field is obtained, if the fringes of W, and W, are exactly superposed in the plane 141. The compensation of the birefringent elements is unusual in this arrangement in not being made in the pupil planes, but in a plane near to the object. A compensated (suppression of diaphragm) interference microscope is also easy to realize in practice for an objective lens operating by [1957] placed a very thin Savart plate against reflection. FRANCON the front element of the objective lens, and SMITH[1950] and NOMARSKI [1955] described an arrangement with only a Wollaston prism (cf. Fig. 4.4). At large magnifications (large apertures) the beam splitters, which are shown in Figs. 4.l.c and d, may be used. Other possibilities are described by NOMARSKI[1955], who modified the Wollaston prism (the axis of the upper wedge forms a small angle with the plate surface) and used a combination of two Wollaston prisms after the microscope objective. Another kind of beam splitter was used by SMITH [1955]. He placed transparent diffraction gratings as beam dividers in the foci of the objective and the condenser, respectively, so arranged that only the zero and first order spectra were transmitted to the image plane. 6 -2.2. W a v efront reversing interferometry

Wavefront reversing interferometers (cf. sec. 2.2.2.2)have also been introduced in microscopy. PHILPOT [19501 describes an arrangement utilizing a prism very similar to Kosters double image prism. 6.2.3. Wavef ront double reversing interferometry

PHILPOT [I9501 used an interferometer, which is shown in Fig. 6.4. Here, the two beams pass in opposite directions around a spiral

path, the specimen, A, being mounted between two cover slips which are placed between two objectives. By means of the two further objectives introduced, 0, and O,, in the cyclic path, an image of the object under test is formed there and thus a double reversing (spirocyclic) interferometer is obtained. A shear is introduced by inclination of one of the mirrors.

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6.3. LONGITUDINAL SHEARING INTERFEROMETRY

In the methods using longitudinal shearing, the two images are doubled in the direction of the axis of the microscope. The focusing is made on one of the images, and, because of the smalI depth of focus of the microscope, the other image is out of focus, but it still gives a practically uniform coherent background, so that it can interfere with the first image. Longitudinal shearing is mostly formed by an uniaxial crystal plate, cemented to the first lens of the objective. The birefringent plate can be cut in any way. SMITH[1955] constructed an interference microscope based on this

Fig. 6.4. A double reversing interference microscope with a triangular path.

Fig. 6.5. Jamin type longitudinal shearing interference microscope using two crystal plates with opposed birefringence.

principle, which is shown in Fig. 6.5. A small birefringent plane parallel plate, cut parallel to its axis, is mounted against the microscope objective 0. A compensation system (suppression of any diaphragm), consisting of a second crystal plate - the birefringence of which has an opposed sense in respect to the first plate - is cemented to the condenser C (the axes of the two plates are parallel). Under these conditions the wave which is extraordinary in the first plate is also extraordinary in the second, but in an opposite sense. Since, however, most available crystals, such as quartz and calcite, which have opposite signs of birefringence usually also differ in their refractive

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properties, precise compensation is not achieved. I n practice this can be overcome by obtaining the small amount of remaining compensation from a birefringent lens located at the first focal plane of the condenser (see Fig. 6.5). Fig. 6.6 shows a similar arrangement described by Franqon (GORLICH [1962]). The two identical crystal plates (cut obliquely to the axes) mounted to the objective 0, and the condenser C, respectively, ,form, together with a 112 plate, a Jamin type interferometer. A Wollaston prism, W, in the focal plane of the condenser gives a uniform field.

\

0

I

Fig. 6.6. Jamin type longitudinal shearing interference microscope using two identical crystal plates.

Fig. 6.7. Double-focus interlerence microscope using two birefringent lenses.

SMITH[1950, 19551 and Philpot (FRANCON [1952b]) have developed a double focus system, consisting of two cemented birefringent lenses, shown in Fig. 6.7. The convex lenses are cut perpendicularly and the concave lenses parallel to the optical axes; the axes of the concave lenses are perpendicular with respect to one another. The outer faces of the birefringent combination are parallel, and thus it only exerts a lens effect on the extraordinary rays. The two birefringent lenses must be in conjugate positions, preferably at the front focal surface of the condenser, C, and the rear focal surface of the objective, 0,respectively. For objectives of higher powers, these surfaces are within the lens

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combination of the objective and it becomes necessary to use a special objectivc having a birefringent lens component. The same effect can also be obtained by a cyclic system. PHILPOT [1950] introduced the system shown in Fig. 6.8. Here, the two initially divided beams pass in opposite directions around a triangular path, the object, A, being mounted between two identical objcctives on a common axis. The required beam separation in the object space between the objectives is achieved by slightly defocusing one of the objectives (e.g., by moving the two objectives and the object under test situated betwcen them out from the middle point of the two \

'.

Fig. 6 . 8 . 1)oublc-focus triangular path interference microscope.

symmetrically placed mirrors). The beam which leaves the microscope system through the focused objective provides a correctly focused image of thc object, but the other bcam, leaving through the defocused objective, provides a superposed out-of-focus image of the object, Consequently, the final image, A', presents an interferometric comparison bctween a true image of the object and a smoothed defocused image of it. 6.4. MEASUREMENT O F OPTICAL PATH DIFFERENCE

The 'half shadow' mcthod is mostly used in the case of instruments not using polarized light (in these instruments thc light can be

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polarized in an arbitrary direction in order to measure birefringent objects). The measurement of the path difference is here performed in the following way: A small reference spot of light introduced in the center of the field of view is set to disappear against the object t o be measured and against the background by operation of the path difference control. The difference between the two settings with an appropriate constant gives then the optical thickness. In the polarizing interference microscope the measurements can be made in a particularly convenient manner. Here the vibrations of the sheared beams combine to form plane polarized light with an azimuth depending on the path difference between them. The problem is here resolved into that of determining this azimuth. I n the eyepiece interferometer of Franson (see Fig. 6.2), optical path differences can be measured very easily and accurately (precision better than f l 0 A) especially by means of a direct reading compensator device designed by Johansson (cf. sec. 6.2.1). In the methods described in sec. 6.2, allowing direct recording of gradients (phase uniform field of view is crossed by fringes), measurement of the optical thickness of isolated details can be made directly from the fringe displacement, by using strong doubling.

References BAKER, L. R., 1955, Proc. Phys. SOC.B68, 871. BARRELL, H., 1949, Nature 164, 599. BARTHOLOMEYCZYK, w., 1960, 2. Instrum. 68, 208. BARTHOLOMEYCZYK, W., 1962, 2. Instrum. 70, 212. BATES,W. J., 1947, Proc. Phys. SOC. 59, 940. BEER,A., 1955, Vistas in Astronomy 1 (Pergamon Press, London) p. 377. BROWN, D., 1954, Proc. Phys. SOC.B67, 232. BROWN, D. S., 1955, J. Sci. Instr. 32, 137. BROWN, D. S., 1959, Interferometry, National Physical Laboratory Symposium No. 1 1 (H.M. Stationary Office, London), p. 253. BROWN, D. S., 1962, J. Sci. Instr. 39, 71. BRYNGDAHL, O.,1957, Acta Chem. Scand. 11, 1017. BRYNGDAHL, 0.and s. LJUNGGREN, 1960, J . Phys. Chem. 64, 1264. BRYNGDAHL, O., 1962a, Monatsh. Chem. 93, 984. BKYNGUAHL, O.,1962b, Arkiv Fysik 21, 289. BKYNGDAHL, 0.and S. LJUNGGREN, 1962, Acta Chem. Scand. 16, 2162. BRYNGDAHL, O., 1963, J . Opt. SOC.Am. 53, 571. CATALAN, L. and M. F R A N ~ O 1960, N , Rev. Opt. 39, 1. CHEVALERIAS, R., Y . LATRON and C. VERET,1957, J. Opt. SOC.Am. 47, 703. DONATH, E. and W. CARLOUGH, 1963, J . Opt. SOC.Am. 53, 395. DREW,R. L., 1951, Proc. Phys. SOC. B64, 1005.

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DYSON, J., 1957, J. Opt. SOC.Am. 47, 386. DYSON,J., 1960, J . Opt. Soc. Am. 50, 754. DYSON,J., 1963, Optica Acta 10, 171. ERICSSON, J. and L. P. SJOFALL,1960, Optica Acta 7, 105. FLUGGE, S., 1956, Encyclopedia of Physics (Springer Verlag, Heidelberg) 24, p. 171. FI~ANCON, M., 1952a, Rev. Opt. 31, 65. FRANCON, M., 1952b, Contraste de phase et contraste par interfkrences (Hev. Opt., Paris) p. 42. F R A N ~M. N and , R. SICKGENT, 1955, Optica Acta 2, 182. FRANCON, M., 1957, J . Opt. SOC.Am. 47, 528. FRANCON, M. and Y. GANUON,1958, Optica Acta 5, 78. FRANCON, M. and T. YAMAMOTO, 1962, Optica Acta 9, 395. GARTNER, W., 1958, Optik 25, 281. GATES,J . W., 1955, Proc. Phys. Soc. €368, 1065. GIRARD, A., R. LENFANT and N. LOUISNARD, 1959, Recherche akronautique 72, 35. GORLICH,P., 1962, Optik und Spektroskopie aller Wellenlangen (Akademie Verlag, nerlin) p. 195. GUILD,J., 1957, Phys. Soc. Year Book, p. 30. HARIHARAN, P. and D. SEN, 1959, J . Opt. SOC.Am. 49, 1105. HARIIIARAN, P. and R. G. SINGH,1959, J . Opt. SOC.Am. 49, 732. HARIHARAN, P. and D. SEN, 1960a, J . Sci. Instr. 37, 374. HAKIHARAN, 1’. and D. SEN, 1960b, Proc. Phys. Soc. 75, 434. HARIHARAN, P. and D. SEN, 1961, J. Sci. Instr. 38, 428. HARIHARAN, P. and I). SEN, 1962, Optica Acta 9, 159. HOPKINS, H. H., 1950, Wave Thcory of Aberrations (Clarendon Press, Oxford) p. 48. HOPICINS, H. H., 1955, Optica Acta 2, 23. INGELSTAM, E., 1955, Arkiv Pysik 9 , 197. INGELSTAM, E., 1957, J . Opt. SOC. Am. 47, 536. JOHANSSON, L. P. and B. M. AFZELIUS, 1956, Nature 178, 137. KELSALL, D., 1959, Proc. Phys. Soc. 73, 465. LEBEDEFF,A.-A,, 1930, Rev. Opt. 9 , 385. LENOUVEL, L. and F. LENOUVEL, 1938, Rev. Opt. 17, 350. LOFIMANN, A., 1954, Optik 11, 478. MOLLIST,P.,1960, Optics in Metrology (Pergamon Press, Oxford) p. 227. MONTGOMERY, A. J., 1964, J. Opt. SOC. Am. 54, 191. NEUGEBALJJ~R, E., 1955, Messung und Sichtbarmachung von milrroskopischen Iliasenobjekten mit Hilfe eines einfachen ’Polarisations-lnterferenzMikroskopcs’, Lecture a t the annual meeting of ‘Verband Deutscher Physikaliacher Gesellschaften’ in Wiesbaden 22-28.9.1955. NOMARSKI, G., 1952, French Patent Specification 1,059,123. NOMARSKI, G., 1955, J. Phys. Rad. 16, 9 S. NOMARSKI, G. and A. R. WEILL, 1955, Rev. Metallurgie 50, 121. PHILUERT, M., 1958, Recherche akronautique 65, 19. PHILPOT, J . ST. L., 1950, U.K. Patent Specification 645,464. RENET,C., 1956, Rev. Opt. 35, 235.

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RONCHI,V., 1923, Annali della R. Scuola Normale sup. Univ. di Pisa 15. RONCHI,V., 1926, Rev. Opt. 5, 441. SAUNDERS, J . B., 1955, J . Opt. SOC.Am. 45, 133. SAUNDERS, J . B., 1957, J . Research NBS 38, 27. SAUNDERS, J . B., 1961, J . Research NBS 65B, 239. SAUNDERS, J. B., 1962, J . Research NBS 66B, 29. SMITH,F. H., 1950, U.K. Patent Specification 639,014. SMITH,F. H., 1955, Research 8 , 385. STEEL,W. H., 1958, Proc. Roy. SOC.A249, 574. TSURUTA, T., 1963a, Appl. Optics 2, 371. TSURUTA, T., 1963b, J. Opt. SOC. Am. 53, 1156. VAN HEEL,A. C . S. and A. WALTHER,1958, Optica Acta 5, 47. WAETZMANN, E., 1912, Ann. Phys. (4) 39, 1042. WEINSTEIN,W., 1953, Nature 172, 461.

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