Axial Twist and Planar Inversion Interferometers

Ronald Alpiar

The theoretical background for a type of common path interferometer, the so-called axial twist interferoin-eter, is discussed. Two devices capable of producing axially twisted bundles of rays are illustrated. Appli-cations of the system are suggested with particular reference to interference microscopy. A simpler, re-lated device, the planar inversion interferometer, is also described, and the characteristics and applicabil-ity of the two instruments are compared.

1. Introduction

This paper discusses the theoretical background,practical implementation, and certain applications ofa new design of common path interferometer. Thebasic theory is easily visualized. Imagine an opticaldevice that will, for any input ray, produce a pair ofoutput rays in a special relationship to each other.This relationship insists that a bodily rotation of one ofthe output rays about some axis will bring it into phys-ical coincidence with its brother. This bodily rotationabout an axis, or axial twist w, is to be the same angle forall possible incoming rays. When two such devices areplaced in alignment, it is possible to arrange for recom-bined rays to interfere constructively. The kernel ofthe affair lies in the fact that this interference phenom-enon is entirely unaffected when any optical system isplaced between the two devices, if, and only if, thatoptical system is symmetric about an optical axis. Thedevice thus acts as null test for alignment and opticalsymmetry. These visual notions are formalized inSec. II.

Section III illustrates two simple devices that arecapable of producing the required twisted bundles ofrays.

Section IV mentions three possible applications ofthis principle: as an optical plumb line; as a null testfor alignment and symmetry; and as applied to interfer-ence microscopy, both in transmitted and reflectedillumination.

In Sec. V a simpler, related device, the planar inver-sion interferometer, is described and its properties arecompared with those of the axial twist instrument.

I. Theoretical Background

The intuitive basis of axial twist interferometry hasalready been outlined. We must now make these

The author is with the Eidgenassiches Institut fur Reaktor-forschung, CH-5303 Wtireiilingen, Switzerland.

Received 2 February 1968.

visual ideas a little more precise. We set up a systemof cylindrical coordinates (Fig. 1). In this system anyline (ray) can be defined by four parameters, r, , a,and d. We now define an w-X congruent pair of rays as apair of rays whose defining parameters satisfy r = r2,al = a2, i1 = 2, 02 = 0-w, and which meet any planeperpendicular to the z azis at equal optical distances(at wavelength X) from a point of common origin.

Remark 1: The congruence of a pair of rays is areflexive property if and only if X is a multiple of 7r.

Remark 2: If ray 2 is rotated bodily about the zaxis through an angle w, it is brought into coincidencewith ray 1. Since the two rays have traveled equaldistances, there will be constructive interference atwavelength X.

Remark 3: To be strictly accurate, one should al-ways speak of congruence with respect to a given axis,since, in general, a pair of rays does not remain con-gruent if the axis is displaced.

Remark 4: If the members of the congruent pair ofrays have traveled equal optical distances from acommon source point, for all values of , we speak of anw congruent pair, omitting the X.

When dealing with sets of congruent rays, membersof the set may be specified by a single parameter 0, theazimuthal angle, since the other three parameters areequal over the set.

Theorem: Let S be an optical reflecting or refract-ing surface. Let any w-X congruent pair of rays beincident on S and result in a pair of output rays (byreflection or refraction). If S is symmetric about theoptic axis, an co-X congruent output pair results fromthe input pair. Conversely, if S is not symmetricabout the optic axis, at least one w-X congruent inputpair of rays will result in a noncongruent output pair.

Proofs: Although the theorem can be proved withmathematical rigor, the proof is algebraically lengthyand unilluminating. Instead, we may resort to asimple Gedankenexperiment. Label the members of theincident pair of rays A and B producing output rays A'

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1461

Fig. 1. Ray specified by cylindrical coordinates r, 0, a, f.

and B'. Given that A and B are a congruent pair and .isymmetric surface, we have to prove that A' and B' arealso congruent. Imagine that the whole system, raysand surfaces, is rotated rigidly through an angle about the optic axis. Tbet A is brought into coinci-dence with B and the new output from ray A, ray A'say, is -X congruent with A' (by rotation). Butsinceit is symmetric about the optical axis, the rotation hasleft the optical surface S entirely unaffected. Thus,since the rotated A coincides with B, the outputs, A"and B', must also coincide. Hence, A' B' form anw--X congruent pair.

Conversely, if S is not symmetric about the axis, wecan always choose a plane perpendicular to the opticalaxis which cuts S in a curve not symmetrical about theoptical axis (that is, a curve that is not a circle centeredat the optical axis). We can further choose a circle,centered about the optical axis, in the cutting plane,that meets the curve in at least two points, the directioncosines of the normal to S at these points being unequal(in cylindrical coordinates). We can now construct anappropriate w-X pair of rays to be incident on the sur-face S at these two points. The incoming pair haveequal direction angles a and . The normals to thesurfaces have unequal direction angles. Hence, theoutput pair of rays must also have unequal directionangles a and 13. Hence they cannot be a congruentpair.

We now define an O-X twister to be an optical devicesuch that each input ray to the device produces alco-X congruent output pair; specifically each member of any set of incoming congruent bundles of rays willproduce two output rays with azimuthal angles - - 0,-y + co - 0; y is an angle dependent only on the orienta-tion of the twister in the coordinate system chosen.

The values of o and X are characteristic of the deviceand the same for all input rays. The optical axis ofoutgoing congruent pairs is also fixed relative to thedevice and is called the optical axis of the twister.Two other directions in a plane perpendicular to theoptical axis are also fixed relative to the twister. Theyare directions with azimuthal angles 1-y and (y + )XIf an input ray enters at one of these positions, one or

other of the output rays will coincide with it. Thesedirections are therefore called the principal axes of thetwister.

As for congruent ray pairs, we may omit the X whenthe device produces w congruent output pairs for all in-plt ays. It is then siml)ly a tister.

Let us now consider what, occurs when two w-Xtwisters are set up with their optical axes coincident andtheir principal axes parallel. Any ray, azimuthalangle 0, entering the first twister, produces two outputrays, angles -y - 0 and -y + - 0. These then enter thesecond twister and produce, in turn, four congruentrays, with azimuthal angles, 0, 0 + a, 0 - co, 0. Since thefirst and fourth rays are coincident and in phase, con-structive interference (at wavelength ) will take place.Constructive interference will also take place betweenthe second and third rays if = r (other multiples of 7rare either degenerate cases, or the same as c = r).

Now place any axially symmetric optical systemwhatever between the two twisters. Since ow-X con-gruent pairs remain so after passing through the sym-metrical optical surfaces, exactly the same interferenceconditions will obtain as when the space between thetwo twisters was empty.

However, any axially asymmetric dioptric placedbetween the twisters will disturb the interference con-dition and give rise to a characteristic interferencepattern.

Ill. Practical Implementation

The simplest device that will act as a twister is illus-trated in Fig. 2. In this diagram, the two inversionprisms are shown separated from the beam splittercube for clarity. In practice they would be cementedto the cube. One prism produces a lateral inversion ofinput rays and the other a vertical inversion. Theresult is a r twister. The output pairs have traveledexactly the same optical distance when t + h1 = t2 + h2.This condition can be realized by small adjustments ofthe relative positions of the two beam splitter prisms,allowing for slight changes in thickness and wedge ofthe cement layer joining them. It has been pointedout to the author that this is rather a difficult opticalfeat, but that an equal phase condition can be attainedby cutting pairs of component prisms from a largerprism (compare Ref. 1). Adjustments of this typecould also produce output rays that are exactly out of

exit face

cube

entry inversionface prism

lateral inversion^iX I7¶/ prism

Fig. 2. Simple r twister.

1462 APPLIED OPTICS / Vol. 7, No. 8 / August 1968

recan.biner.cube

elticaltranslator

exit

iX\II I ~~~late"]- I-----\z- < -- irvers-

Iz X i ar.1 itter

lateral ~\~5J translator \

\ >ev3;9 | vertical

I< - inversion f

Fig. 3. r twister, second design.

phase for a given wavelength, but then the device is nolonger wavelength independent. By appropriate ad-justments, two such twisters could be set up to givedestructive interference of the recombined rays (darkbackground).

Altering the orientation of the two inversion prismswill result in an co twister, co w7r. In particular co can bemade infinitesimally small.

The disadvantage of the device is that the two roofedges of the inversion prisms cut centrally across theaperture. Since these edges cannot in practice begeometric lines, the aperture is obscured by a thincentral cruciform obstruction. This can only beavoided at the cost of constructing the much morecomplicated system illustrated in Fig. 3. In this de-vice a bundle of rays enters the splitter cube from theleft. The transmitted bundle enters the lateral inver-sion prism, is laterally inverted and shifted one unit inthe Y direction, it then enters the vertical translatorand is shifted one unit in the Z direction and then entersthe recombiner cube. The reflected bundle enters thevertical inversion prism, undergoes vertical inversion,and is shifted one unit in the Z direction; it then entersthe lateral translator and is there shifted minus oneunit in the X direction, finally entering the recombinercube. In practice it is doubtful if such an elaboratesystem could be set up to the necessary accuracy.

In either case, the relative phases of the two beamscould be controlled optically by introducing crossedpolarizing filters between the beam splitter cube and theinversion prisms and an orientable retarding plate inthe recombined beam. It has also been pointed out tothe author that an optical wedge would achieve thesame phase control.

IV. Applications

Three applications of a pair of axial twisters come tomind.

(a) The interference effect is sensitive to misalign-ment of the order of a fraction of a wavelength. Inter-ference effects will indeed only be observed at all whenthe alignment is of the order of the coherence lengthof the light source used. Thus, if two twisters can bealigned to within the said coherence length by means of

less sensitive physical optics instrumentation, thetwisters can be readjusted to a far greater degree ofalignment by observing interference fringes. Thesystem can thus serve as a supersensitive optical plumbline.

(b) A pair of properly aligned twisters will give aninterferometric null test for the centration of opticalelements and systems placed between them. This nulltest is completely unaffected by any aberrations of theoptics, other than those caused by misalignment. Thispoint is worth stressing by comparison of the presentsystem with that described by Murty and Hagerott. 2

In the latter system, one beam is rotated an angle 0relative to the other and the two beams are then super-imposed and interference effects are observed. As theabove authors point out, their system is then sensitiveto unsymmetrical aberrations (coma, astigmatism) ofthe optics being tested. Contrast the case of a pair oftwisters; the two beams are rotated some angle to eachother, pass through the optics being tested, and arethen rotated back to their original orientation by thesecond twister. The result is that even the unsym-metrical aberrations do not affect the centration nulltest. For, in the end, one is interferometrically com-paring pairs of rays which, if they have undergonecomatic or astigmatic aberration, have suffered thesedistortions in exactly the same degree. The secondtwister perfectly compensates the action of the first,and incidentally thus permits one to use a source offinite size.

(c) A pair of twisters can be placed above the objec-tive and below the condenser of a conventional micro-scope. When properly aligned, this will function as aninterference microscope. The system presents thefollowing features.

(1) Being a common path interferometer, no duplica-tion of condenser and objective is necessary.

(2) The interferometer elements are well clear of thecritical region between the condenser and the objective.

(3) Ordinary commercial optics may be used. Thesame pair of twisters will function equally for all com-binations of condenser and objective. Provided thatthe system is nicely aligned, no readjustments arenecessary when focusing or even when changing objec-tives.

(4) The interference pheiiomenon is wavelengthindependent and a white light source may be used.The twisters may be set up to give darkground illu-mination, or any other ground on the Newton colorscale. If polarizing filters are incorporated in thetwisters, continuously variable illumination back-ground may be obtained, as in double refractiveinterference instruments.

(5) When using r twisters, 100% contrast is ob-tainable, since all rays contribute to the interferenceeffect.

(6) The interferometer elements are simple, con-sisting only of reflecting and refracting plane surfaces.

(7) The full aperture of the optics is utilizable with-out loss of contrast.

August 1968 / Vol. 7, No. 8 / APPLIED OPTICS 1463

exit face

beam-splittercube

entry - > inversionface \/ prism

| path lengthcompensator

-o- reflectivesurface

(a)

t exit face

t entry face

(C)(b)

Fig. 4. (a) Iverter device, (b) two components of an inverter, (c) two components of a dire t vision inverter.

The disadvantage of the system lies in the imageduplication produced. Two images of the field of viewand its contents are seen. With maximum twist angle,one field is rotated at an angle r relative to the other.In certain cases this will result in confusion owing toportions of the object under examination overlappingeach other. However, in many cases offensive overlapis avoidable. Nonoverlap can be achieved: (1) for anisolated object whose smallest convex diameter is lessthan half the field of view; (2) for sections of a largeobject lying at distances ot greater than half diameterof the field of view from its edge; and (3) for a distri-bution of objects separated by distances not less thantheir diameters.

Instead of using twisters, we could make use of apair of e twisters, e being a very small angle such thatduplication at the edge of the field is below the limit ofvisual resolution. In this case we obtain differentialinterference effects where the gradients of optical pathin the specimen produce interference coloring. Themaximum contrast falls to 50%, since only two of thefour rays are properly recombined. The system is thencomparable to that of Franqon.3 Frangon uses a pairof compensating birefringent elements that produce atiny lateral shift of pairs of rays. His system thengives interference coloring to gradients in the specimenperpendicular to this lateral shift. Similarily, the pairof e twisters will color interferometrically azimuthalgradients in the specimen, but the sensitivity of thesystem will not be constant over the field of view, beingzero at the center and rising to its maximum value atthe edge of the field of view.

The device can also be applied to microscopy in re-flected light. Here only one twister element, placedabove the objective, is required.

The author should point out that the proposed de-vices have not yet been constructed or experimentallytested and that the above claims are to that extenttentative. It is to be hoped that some optical insti-tute will take up the experimental side of the proposeddevices, as this is beyond the capabilities of the authorhimself.

V. Planar Inversion Interferometry and DevicesIn Sec. II we defined a pair of co-X congruent rays by

reference to rotation about a fixed axis. In this section,we define an inverse pair of rays by the requirementthat one ray is the reflection of its brother in a fixedplane. This fixed plane will be called the plane of in-version. Analogously, we can speak of a X inversepair, as having traveled the same optical distance, atwavelength X from a point of common origin.

An inverse pair of rays, when encountering any opticalsurface symmetrical about the plane of inversion willgive rise to another pair of inverse rays. This may beproved in a manner similar to the proof in Sec. II. Fur-ther, we may define a X inverter as a device that willproduce an inverse pair for each ray that enters it.The plane of inversion is fixed relative to the X inverter.

If two inverters are placed with their planes of inver-sion coplanar, constructive interference of the recom-bined rays take place. Any movement of either in-verter that leaves their planes of inversion coplanar willhave no effect upon the interference conditions. Smallmovements that separate the two planes produce char-acteristic interference patterns. If any optical surfaceis placed between the two coplanar inverters, interfer-ence conditions will be unaffected, provided that thesurface is symmetric about the plane of inversion.

Figure 4(a) illustrates a simple inverter device.Most of the remarks in Sec. III apply equally to theinverter. However, the field of view, in this case, isdisturbed by only one roof edge.

Technically, the inverter is simpler to construct thaiithe twister. In either case, the devices can be madefrom two pieces of glass cemented together. But thetwister requires a pair of roof faces to be constructed onone of its components. These roof faces disappear inthe inverter device. Figure 4(b) shows the shapes ofthe two components of an inverter. As shown in Fig.4(c), a simple modification of the forms of the compo-nents will yield a direct vision instrument.

In application, inversion interferometry is perhapsnot as useful as twist interferometry. The alignmentdevice is insensitive to movements within the inversion

1464 APPLIED OPTICS / Vol. 7, No. 8 / August 1968

(a) (b)

Fig. 5(a) and (b). Image doubling in the fields of view of theI axial twist and the planar inversion interference microscopes.

plane. One can, however, conceive that two pairs ofinverters, operating in orthogonal planes might be ofuse if one required independent measures of displace-ment in two directions.

The inverter device is quite similar to Koster's doubleprism.4 There is, however, one important difference.In the inverter the inverted pairs of beams are recom-bined within the device itself, before passing out to theoptical system under test. The inverter is thus a truecommon path interferometer in which each of the com-ponent beams separately fills the entire aperture of theoptics being tested. This is important, e.g., in mi-

croscopy, where a sacrifice of resolving power is un-desirable.

As a centration device, a pair of coplanar twisterswould only be of use in aligning cylindrical optics.

The application to microscopy carries over from thetwist to the inversion interferometers practically un-changed. The image doubling produced by the twoinstruments is illustrated in Fig. 5(a) and (b). Theinversion interferometer is possibly easier to use inpractice since it requires only one degree of centrationof the microscope optics.

The author would like to thank the Directors of theSwiss Federal Institute for Reactor Research for sup-porting the publication of this paper.

References1. J. B. Saunders, Appl. Opt. 6, 1581 (1967).2. M. V. R. K. Murty and E. C. Hagerott, Appl. Opt. 5, 615

(1966).3. M. Frangon, in Progress in Microscopy (Pergamon Press,

New York, 1961), pp. 107-109, 113-117.4. W. Kosters, Interferenzdoppelprisma fir Messzwecke, Ger-

man Pat. 595211 (1934).

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GARDNER LABORATORY, INC., 5521 Landy Lane, Bethesda, Maryland20014, was established in 1924 as a consulting and testing laboratory forchemical, physical, and appearance measurements. It has developedinstruments for measuring appearance of materials in controlling qualityof completed products, as well as high-speed color sorting equipment,optical-surface-finish evaluators, and color and gloss instrumentation.

GCA CORPORATION, Burlington Rd, Bedford, Massachusetts 01730,manufactures and develops precision instruments and industrial processequipment and conducts research into the physics of the earth's atmosphereand space and planetary environments. Operating elements of the com-pany are: the GCA Technology Division for advanced research associatedwith military and space programs; the David IV. Mann Co. producingprecision photographic analysis equipment, photom'sk production systems,and lens test and camera calibration instruments; Vacuum Industries, Inc.producing vacuum and high-temperature industrial processing systemsand vacuum instruments; Precision Scientific Co. mnufacturing appa-ratus and instruments used for measuring, testing and control in labora-tories, plus process control monitoring and analysis instruments for petro-leum industry; and McPherson Instrument Corp. manufacturing spec-trometers for research and analytical work in the soft x-ray to near ir regions.

GENERAL ANILINE & FILM CORPORATION, PHOTO & REPRO DI-VISION, 140 W. 51st St, New York, New York 10020.

GENERAL ELECTRIC COMPANY, LAMP DIVISION, Nela Park, Cleve-land, Ohio 44112, is headquarters for forty plants, seventy sales and dis-tribution offices, and six product departments, all involved in the manu-facture and distribution of large lamps, miniature lamps, photo lamps,lamp glass, and lamp metals and components; it also directs the operationsof ten engineering and research laboratories. The Nela Park facilityincludes research and development laboratories, with interests in lampresearch, lamp engineering, radiant energy effects, light sources and theirproperties, spectra, color and color systems, optical properties of phosphors,etc.

GENERAL TELEPHONE AND ELECTRONICS LABORATORIES, INC.,Bayside, New York 11360, carries out a research program for its parentcompany, General Telephone and Electronics Corporation and its manu-facturing subsidiaries, Sylvania Electric Products Inc., Automatic ElectricCompany, and Lenkurt Electric Co., Inc., covering such areas as com-munications, film electronics, energy storage and conversion, electronicdevices, electronic materials, material analysis, and theoretical studies.

GOERZ OPTICAL COMPANY, INC., 461 Doughty Blvd, Inwood,New York 11696, founded in 1895, manufactures optical products, includ-ing a variety of lenses and other products, such as viewfinders, aircraft at-tack simulators, image reversers, condenser systems, high-speed shutters, andtracking systems for missile flight and reentry recording. In addition, itdesigns and manufactures optical and electrooptical instrumentation tospecification.

GRIMES MANUFACTURING COMPANY, 515 N. Russell St, Urbana,Ohio 43078.

GRUMMAN AIRCRAFT ENGINEERING CORPORATION, Bethpage,New York 11714, is engaged in the development and production of aircraft,space, and ocean systems bringing to bear all forms of technology to meet therequirements of defense, scientific exploration, and economic growth. Opticsfinds expression in Grumman's major programs for space astroseosny,manned lunar exploration, and airborne weapons systems. The opticaleffort ranges from basic research to direct applications, including investiga-tions of new techniques and development of electrooptical prototype systems.

W. & L. E. GURLEY, 514 Fulton St, Troy, New York 12181, has manu-factured engineering, surveying, and other scientific instruments since itsfounding in 1845. It produces standard optical instruments, as well asoptical systems designed to specification. Among its products are high-accuracy linear and circular scales on glass in combination with opticalcoincidence reading systems and precision resolution targets and reticlesfor optical and photoelectric instruments made by mechanical, photographic,electroforming and electroetching methods.

THE HARSHAW CHEMICAL COMPANY, DIVISION OF KEWANEEOIL COMPANY, 1945 E. 97th St, Cleveland, Ohio 44106, basically amanufacturer of industrial chemicals, produces and markets through itsCrystal-Solid State Department a complete line of optical crystals, micro-wave materials, radiation detectors and associated nuclear electronics. Someapplications include ir, uv, and military optics, x-ray diffraction, nuclearphysics, medical research, solid state research and thermoluminescencedosimetry.

HILGER AND WATTS, INC., 8035 Austin Ave, Morton Grove, Illinois60053, was organized in 1959 by Hilger and Watts, Ltd., London, makers ofscientific instruments, jointly with Engis Equipment Company of MortonGrove, to maintain technical liaison and to share research training and con-tinuing studies of future needs i the field of scientific instrumentation.Personnel and consultants are available in the fields of vacuum spectros-copy, interferometry, x rays, engineering optics, and precision technicalinstrumentation.

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1466 APPLIED OPTICS / Vol. 7, No. 8 / August 1958