6
Stereoscopic Imagery in a Type of Stereoscopic Microscope A. J. Kavanagh An analysis is made, from the point of view of first order geometrical optics, of the stereoscopic imagery in the type of stereoscopic microscope which consists of a simple magnifier followed by a sort of binocular field glass. The imagery cannot be orthostereoscopic. Equations are given for the transformation from object space coordinates to image space coordinates, for lateral and longitudinal magnifications, for the ratio of longitudinal to lateral magnification, and for the factor by which the instrument enhances the ability to detect small differences in depths of the object. When the eyepiece axes are not parallel, certain com- plications in interpreting the imagery are encountered. Introduction A stereoscopic microscope is one whose construction is such as to provide the observer's eyes with slightly differing views, so as to permit the perception of depth effects. In recent years, a form of stereoscopic micro- scope has become popular which consists in essence of a low power magnifier which forms a virtual (three- dimensional) image of the object, followed by a sort of binocular field glass which views the image formed by the magnifier. The stereoscopic imagery in such microscopes is sufficiently different from that of com- mon earlier types to make it worthwhile to present an analysis of it. The analysis is limited to considerations of geo- metrical optics. It is well known that several factors are involved in making judgments of depth and shape from visual clues, and that the geometry of the image presented to the eyes is only one such factor. (For summaries, see Judge' and von Gruber.2) Neverthe- less, it is an important one, and the treatment in the present paper is limited to it. Expressions such as shape and depth are to be understood to refer to the geometry of the image, abstracted from the other fac- tors influencing perception. Further, the treatment uses only first order optics. The presence of aberrations in the optical system will affect the shape of the geometrical image. Such effects are ignored here. Two questions are of special interest. One is that of the shape of the image as compared with that of the object. What is commonly considered correct imagery is that in which the image is similar to the object in the sense of elementary geometry, with corresponding The author is with the American Optical Corporation, P.O. Box 187, Framingham, Massachusetts 01701. Received 8 November 1968. angles equal and corresponding sides proportional. Thus an object which is a cube will be represented by an image which is also a cube, ordinarily of different size. Such imagery is frequently called orthostereoscopic. One may ask what design parameters of the instrument will produce orthostereoscopic images, or, more gen- erally, how the shape of the image depends on the designof the instrument. The other question is that of how much the instru- ment enhances the observer's ability to detect small differences in depth in the object. This is related to the change in binocular parallax produced, a matter which is discussed below. The Microscope System The layout of a microscope system of this sort is shown schematically in Fig. 1. The magnifier has an equivalent focal length f. Its principal foci are at F and F', respectively. The two viewing systems form- ing the binocular field glass have their objective axes parallel to the axis of the magnifier, and located at equal distances a on either side of it. The entrance pupils of these systems are shown at PL and PR, respectively. The distance from the principal focus F' to the line joining PL and PR is p. (According to the sign convention to be used, p as shown in the figure is negative.) The exit pupils of the systems are PL' and PR', respectively. Their distances from the line of the axis of the magnifier are equal to b, which is usually, if not always, greater than a. (Of course, 2b is equal to the observer's interpupillary distance.) The eyepiece axes are assumed, for the present, to be parallel to the objective axes, as shown in the figure. The case of nonparallelism is discussed later. The two viewing systems are erecting. They are also usually designed to be afocal. However, afocality is not necessary for the present analysis, and no assump- tion will be made as to whether the systems are afocal or not. May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 913

Stereoscopic Imagery in a Type of Stereoscopic Microscope

  • Upload
    a-j

  • View
    214

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Stereoscopic Imagery in a Type of Stereoscopic Microscope

Stereoscopic Imagery in a Type of Stereoscopic Microscope

A. J. Kavanagh

An analysis is made, from the point of view of first order geometrical optics, of the stereoscopic imagery inthe type of stereoscopic microscope which consists of a simple magnifier followed by a sort of binocular fieldglass. The imagery cannot be orthostereoscopic. Equations are given for the transformation from objectspace coordinates to image space coordinates, for lateral and longitudinal magnifications, for the ratio oflongitudinal to lateral magnification, and for the factor by which the instrument enhances the ability todetect small differences in depths of the object. When the eyepiece axes are not parallel, certain com-plications in interpreting the imagery are encountered.

Introduction

A stereoscopic microscope is one whose constructionis such as to provide the observer's eyes with slightlydiffering views, so as to permit the perception of deptheffects. In recent years, a form of stereoscopic micro-scope has become popular which consists in essence of alow power magnifier which forms a virtual (three-dimensional) image of the object, followed by a sort ofbinocular field glass which views the image formed bythe magnifier. The stereoscopic imagery in suchmicroscopes is sufficiently different from that of com-mon earlier types to make it worthwhile to present ananalysis of it.

The analysis is limited to considerations of geo-metrical optics. It is well known that several factorsare involved in making judgments of depth and shapefrom visual clues, and that the geometry of the imagepresented to the eyes is only one such factor. (Forsummaries, see Judge' and von Gruber.2) Neverthe-less, it is an important one, and the treatment in thepresent paper is limited to it. Expressions such asshape and depth are to be understood to refer to thegeometry of the image, abstracted from the other fac-tors influencing perception.

Further, the treatment uses only first order optics.The presence of aberrations in the optical system willaffect the shape of the geometrical image. Such effectsare ignored here.

Two questions are of special interest. One is thatof the shape of the image as compared with that of theobject. What is commonly considered correct imageryis that in which the image is similar to the object in thesense of elementary geometry, with corresponding

The author is with the American Optical Corporation, P.O.Box 187, Framingham, Massachusetts 01701.

Received 8 November 1968.

angles equal and corresponding sides proportional.Thus an object which is a cube will be represented by animage which is also a cube, ordinarily of different size.Such imagery is frequently called orthostereoscopic.One may ask what design parameters of the instrumentwill produce orthostereoscopic images, or, more gen-erally, how the shape of the image depends on thedesign of the instrument.

The other question is that of how much the instru-ment enhances the observer's ability to detect smalldifferences in depth in the object. This is related tothe change in binocular parallax produced, a matterwhich is discussed below.

The Microscope SystemThe layout of a microscope system of this sort is

shown schematically in Fig. 1. The magnifier has anequivalent focal length f. Its principal foci are at Fand F', respectively. The two viewing systems form-ing the binocular field glass have their objective axesparallel to the axis of the magnifier, and located atequal distances a on either side of it. The entrancepupils of these systems are shown at PL and PR,respectively. The distance from the principal focusF' to the line joining PL and PR is p. (According to thesign convention to be used, p as shown in the figure isnegative.) The exit pupils of the systems are PL'and PR', respectively. Their distances from the line ofthe axis of the magnifier are equal to b, which is usually,if not always, greater than a. (Of course, 2b is equal tothe observer's interpupillary distance.) The eyepieceaxes are assumed, for the present, to be parallel to theobjective axes, as shown in the figure. The case ofnonparallelism is discussed later.

The two viewing systems are erecting. They arealso usually designed to be afocal. However, afocalityis not necessary for the present analysis, and no assump-tion will be made as to whether the systems are afocalor not.

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 913

Page 2: Stereoscopic Imagery in a Type of Stereoscopic Microscope

P L _ b -'P.

3b --- (Pk

p

(Negaivees shown)

MagnifierEFL I

Y

Fig. 1. Schematic layout of a stereoscopic microscope of thetype discussed. F and F' are the principal foci of the mag-nifier. PL and PR are the entrance pupil points of the partsof the binocular system. PL' and PR' are the exit pupil points.The optical axes of the two systems at the exit pupils are parallelto each other. The distance 2b is the user's interpupillary dis-

tance.

The Magnifier Imagery

In describing the imagery produced by the magnifieralone, it is convenient to take the origin for coordinateaxes in the object space at F, and that in the imagespace at F'. In both spaces let the positive x axis be tothe right in the plane of the figure, the y axis lie upwardalong the optical axis, and the z axis be perpendicular tothe plane of the figure. Let the subscript 1 denotecoordinates in the object space, and the subscript 2those in the image space. Then we have the well-known relations:

X2 = fX11Yl,

(1)Y2 = _f2/yl,

Z2= fz,/y,.

Without resorting to the use of the equations, it iseasy to see that the imagery produced by the magnifieris not orthostereoscopic. From the definition of theprincipal focus, all lines in the object space parallel tothe optical axis are transformed into lines which, ifextended, pass through F' in the image space. Then, ifthe object is a cube standing on one face on the stageof the microscope, so that its four vertical edges areparallel to the optical axis, the images of these edges,extended, must pass through F', and the image of thewhole cube is a truncated pyramid. If the base of thecube happens to lie in the plane containing F, the baseof the truncated pyramid is at infinity, while its upperface is at a finite distance. Thus, the image is notsimilar to the object in the sense of elementary geom-etry.

The StereomodelThe concept of the imagery produced by the binocular

system is more complex than that of the imagery pro-duced by the magnifier alone. The latter produces apoint image of each object point, and there is noambiguity as to the locations of these image points inspace. They are given by Eq. (1). Now, each sideof the binocular system produces its own image of eachpoint in the magnifier image. We may call these thetwo monocular images. However, the two monocularimages of a given point ordinarily do not coincide inspace; they do not then by themselves determine thelocation of the stereoscopic image of the point. It isnecessary to adopt the point of view common in theanalysis of stereoscopic effects and summarized in thereferences already given. Briefly, the ray joining thecenter of either exit pupil and the corresponding mon-ocular image point determines the line of sight for thatimage point in that half of the system. The inter-section of the two corresponding lines of sight (if theydo intersect) is taken to be the stereoscopic imagepoint of the original object point. The figure formedby the totality of such image points is called thestereoscopic image of the object. The terms stereo-scopic model and stereomodel are also used.

It is the geometry of the stereomodel that is to beinvestigated.

It should be noted that in practice the two monocularimage points must be close enough to the correspondingpoint of the stereomodel so that eyestrain is not intro-duced by the abnormal accommodation-convergencerelationship.

Transformation by the Binocular SystemReturning to Eq. (1), it is convenient to shift the

origin from F' in Fig. 1 to P, the intersection of theline PRPL with the axis of the magnifier. Let F'P = P.the sign convention being chosen so that p as shown isnegative. Let (2,gi2,22) be the coordinates so trans-formed. Then,

X2 = X2 = fx1/yt,

92 = Y2 P = -f2/ -P p, (2)

52 = Z2 = fzi/Yi.

914 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

Page 3: Stereoscopic Imagery in a Type of Stereoscopic Microscope

As a preliminary to determining the directions of thelines of sight at the exit pupils, consider the correspond-ing lines of sight at the entrance pupils.

The direction of the chief ray between the entrancepupil point PR and any point (2, 2,22) can be charac-terized by the unit vector (R,mR,nn) lying along theray, where

1R = (2 - a)/rN,

mR = Y2/rR,

nR = 2/rn,

(3)

andrR [(r2 - a)2 + y22 + Z22]2.

Let M be the angular magnification which the right-hand system produces in the chief ray bundle. (Thisis, of course, equal to the reciprocal of the pupil mag-nification. If the system is afocal, it is also the powerof the telescope.)

Let the unit vector lying along the transformed raythrough PR' be denoted by (',mR,nR). Then it canbe shown by straightforward procedures that:

P R = MlR/DR,

m R = mR/DR, (4)

n'n = Mnn/DR,

where

DR = (MR2 - M2mR2 + M2)k

Using similar notation for the chief ray through theleft-hand system, we have:

IL = (2 + a)/LN,mL = 2/rL, (5)nL = Z2/rL,

whererL = [(?2 + a)2 + g2

2 + z22].

The equations for the transfer from (L,ML,nL) to(lL',mL',nL') are identical with Eq. (4), except for thereplacement of subscripts R by L.

We now wish to write down the expressions for thecoordinates of the image point (,y 3 ,z2) in terms of(1R',mg',nR') and (lL',mL',nL'), the origin being at P'.

In general, for two rays through PL' and PR',respectively, to intersect, they must be coplanar withthe line PL'PR'. That is, using determinant notation,it is necessary that:

1 0 0IR' mR nR' = 0. (6)it mL' nI

It can be shown by substitution from the earlier equa-tions that, since (R,mR,nR) and (L,mL,nL) are co-planar with PLPR, Eq. (6) is satisfied.

Then it can be shown that:

where

G IL'mR - R 'mL'-

Upon substitution from Eqs. (4), (3), and (5), we have:

X3 = b&2/a,

Y3 = b 2/aM, (8)

z3 = b2/a.

This is the transformation by the binocular systemalone.

Imagery of the Whole Microscope

Upon substitution from Eq. (2), we finally obtain

X3 = bfxi/ayi,

y3 = -bf 2/aMyi - bp/aM, (9)

Z3 = bfzilayi.

These equations give the transformation from the objectto the stereomodel.

The Image Shape

Several characteristics of the imagery are worthnoticing.

Planes in the object space which are perpendicular tothe y axis (planes yi = constant) are transformed intoplanes in the stereomodel which are also perpendicularto this axis. (The plane y, = 0 is projected into a planeat infinity.)

Just as lines in the object space parallel to the y, axiswere transformed by the magnifier into lines passingthrough F', so these lines are transformed by Eqs. (9)into lines passing through a common point in thestereomodel space. For, consider an object-space line:x = c; z = d, where c and d are constant. Uponsubstitution in Eqs. (9) and elimination of yi, we havefor the equations of the edge in image space:

x3/c = - (Ay3 + pb/a)/f = z3/d.

These equations are satisfied by (0, -bp/aM, 0), what-ever the the values of c and d. That is, all lines in theobject space parallel to the axis of the magnifier inter-sect at a finite point in the image space. This point is,as might be expected, the stereoscopic image of F'.The image of a cube is then a truncated pyramid, muchas it was in the space immediately following the mag-nifier.

It follows that the imagery in a microscope of thiskind cannot be orthostereoscopic.

The lateral magnification in a plane perpendicular tothe y axis is by definition x3/xl or Z3/Zl Evidently,

X31X1 = Z3/Zl = fb/ayi. (10)

X = b(lL'mR' + IR'mL')/G,

Y3 = 2bmLmMR'/G,

Z3 = 2bmR'nL'/G,

Then the lateral magnification is inversely propor-tional to the distance of the object plane from F.

(7) The longitudinal magnification is by definition dy3/dyl. From Eq. (9) it follows that

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 915

Page 4: Stereoscopic Imagery in a Type of Stereoscopic Microscope

Fig. 2. Optical

dA/dy = 2aM/f2. (13)

Here the factor 2a/f is in effect the angle of convergenceof the projected binocular axes on the object point(strictly, twice the tangent of angle C of Fig. 1), andM/f is proportional to the magnifying power of theinstrument as conventionally expressed.

It is instructive to rewrite this result as follows. LetAt be the total nominal magnifying power of theinstrument. Then, if f is expressed in millimeters, T= 950i11/f, and, when y, = 0,

dA/dy = (2a/f) X (/250).

axes at the exit pupils in a microscopeconverging eyepiece axes.

dy3/dy, = fb/Aay,2.

This may be written

dy3/dy = (a/lMb)(fb/ayi)'.

The comparison is not usually made, but it is in-structive to compare this result with the depth percep-tion available to the user without the microscope.

with Using the conventional 250-m viewing distance and anominal interpupillary distance of 63 mm, the rate ofchange of convergence angle with respect to depth isvery close to 1: 1000. If we define Ef, the enhancementof depth perception, to be the factor by which theinstrument increases the rate of change of convergenceangle, we have, when Yi = 0,

\1il/}

Then the longitudinal magnification is proportional tothe square of the lateral magnification. (This differsfrom the similar relation in the case of the imagery of asingle lens, in that the constant of proportionality a/lblis not ordinarily unity.)

The ratio of longitudinal to lateral magnification isof some interest in considering the character of theimagery in the neighborhood of a point. From Eqs.(10) and (11), it follows that the ratio is f/lilMyi. Thenit is unity (that is, correct in the sense of orthostereo-scopic imagery) only when

Y1 = fbI!. (12)

When the object point is at F, so that y1 = 0, the ratiois infinite. Since the instruments are usually designedso that the object must be near F, to be within theusable depth of field, it is evident that in practice theratio is always greater than unity.

Enhancement of Depth PerceptionWe now consider the effect of the properties of the

stereomodel on the user's ability to detect small differ-ences in depth of the object. The ability to detect suchdifferences is believed to be limited by the observer'sability to sense small differences in the binocularparallax, the angle of convergence of the lines of sightfrom two eyes, as he compares one point with another.Hence, we inquire how this angle depends on the posi-tion of the point in the object space. For simplicity,consideration will be restricted to object points lyingon the optical axis of the magnifier.

Let A be the angle of convergence. Then, A = -2arctan (b/y3). It follows that

dA/dy = 2aMf'/[(f' + pyd)' + aM'yi'J.

We examine only the case yi = 0, since the instrumentsare usually designed so that the object is placed near thepoint F. At this point,

E = lOOOdA/dy = SI/f.

No formal standard seems to have been set up as tothe best value of F at any given power AT. A plausible,but by no means necessary, hypothesis is that E should-equal AT; that if, for example, -l = 10, then E shouldalso be 10. This implies 8a/f = 1, or that the con-vergence angle of the binocular axes in the space pre-ceding the magnifier should be about 142o, that is,about the same as the angle for eyes without the micro-scope, viewing at the standard distance of 250 mm.

It should be evident that the foregoing analysis re-mains valid if, as is usually the case with prismaticinstruments, the eyepieces axes are displaced forwardor backward from the plane of Fig. 1, or are so arrangedas to permit the observer to look at an inclined anglerather than straight downward, provided the eyepieceaxes remain parallel to each other.

Systems with Converging Eyepieces

For increased ease of viewing, some users prefer aninstrument that permits the eyes to converge. To thisend, instruments are made in which the eyepiece axesare at an angle with the plane of symmetry of the instru-ment. The axes of the binocular systems at PL and PRretain their original relation to the axis of the magni-fier.

As shown in Fig. 2, let w be the angle that each eye-piece axis makes with the y3 axis. Then it is sufficientfor present purposes to consider the bundle of chief raysat either exit pupil as being the bundle treated in thecase of parallel axes, but rotated about the axis (0,0,1)through an angle -w in the case of the right-handsystem, and an angle w in the left-hand system.

Using double primes to designate components of thevectors after the rotations,

1R' = IR' cosw + mR' sinw,

(16)MIR' = -' SEW + 'IR' COSW,nitR = ni',

916 APPLIED OPTICS / Vol. 8, No. 5 / May 1969

(14)

(15)

I

. L_III

PL

1__41 .

I

II

Pk

l

Page 5: Stereoscopic Imagery in a Type of Stereoscopic Microscope

and

IL = L coSw - mL' SiTlW,

mL = L' sinw + mL' cosw,

nL" = nL'.

On investigating the coplanarity condition [compareEq. (6) , it turns out that a pair of corresponding raysare coplanar with PL'PR' if and only if the object pointlies in the plane z1 = 0 or in the plane x1 = 0; that is,either in the plane of the system as shown in Fig. 1,or in what might be called the median plane of thesystem, perpendicular to it.

For all other points, what is known as verticalimbalance is introduced. That is, for the two eyes tofixate the respective monocular images simultaneously,they must rotate so that their optical axes are no longercoplanar with the line joining their centers of rotation.The eyes have only a small capability for rotations ofthis sort, and the amount of imbalance must therefore bekept small.

Obviously, in such a situation, the definition ofstereoscopic image points given above does not apply,except in the two planes for which corresponding raysdo intersect. However, experience shows that theimpression of a stereoscopic model is received by theobserver if w is only moderately large. It would benecessary to develop a more complex definition of thestereoscopic image to deal completely with this case.We do not attempt this, but limit consideration to thetwo planes in which the simpler definition remainsvalid.

Let ( 4,y4,z4) be the coordinates of a point in thenew stereomodel. These coordinates will be expressedas functions of 3,y3,z3, the coordinates in the stereo-model with parallel eyepiece axes.

Imagery When xl = 0Consider first the midplane of the instrument, for

which xl = 0. By combining Eq. (16) with the equa-tions giving the relations between the lines of sightand the points of the stereomodels, it can be shown that

X4 = 0,

b(b sinw + y3 cosw)Y b cosw - 3 sinw (17)

bz3Z4=

b cosw - y3 sinw

From inspection of the equations, it can be seen thatstraight lines in the earlier model are imaged as straightlines, and those which were perpendicular to the y axisremain perpendicular to it.

The point (0,- bp/aM,0), which is the stereoscopicimage of F', is transformed into a point (0,y4,0) on they axis, where

14 = b(aM sinw -p cosw)/(aM cosw + p sinw). (18)

In principle, 4 could be made infinite by making p =-aMcotw, but, in practice, this would require eitherlarge values of p or large values of w.

The lateral magnification produced by the trans-formation of Eq. (17) is

Z4/Z3 = b/(b cosw -y3 sinw), (19)

and the longitudinal is

dy4 /dy3 = b/(b cosw- y sinw)'. (20)

Hence the ratio of longitudinal magnification to lateralproduced by the transformation is

b/(b cosw - y3 sinw). (21)

By combining this with the value obtained above for thesystem with parallel eyepiece axes, and substituting fory3 from Eq. (9), the ratio for the whole system is seen tobe

af/[yi(aM cosw + p sinw) + f2 sinw].

The ratio equals unity when

yi, = f(a - f sinw)/(aM cosw + p sinw).

(22)

(23)

This result is to be compared with Eq. (12). In thepresent case, it is possible, for example, to make y' = 0by choosing w so that sinw = a/f. From the discussionof the ratio a/f following Eq. (13), it is seen that,neglecting the difference between tangents and sines,this requires that the angle between the eyepieceaxes equal the angle of the projected binocular axes atthe object point F.

Imagery When Zi = 0

Consider now the plane corresponding to zL = 0. Bymaking the appropriate substitutions, it can be shownthat

-2bx3y3X4 = (X32 + y32 - b2) sin2w - 2by3 cos2w

-2b[y32 cosSW - (x32 - b2) sin2w + by3 sin2w]=4 (X32 + y32 - b2) sin2w - 2by3 cos2w

(24)

From inspection of the equations, it can be seen thatstraight lines in the stereomodel with parallel eyepiecesare imaged as curves, not as straight lines, in the presentmodel.

Further, the lateral magnification corresponding toa given value of y3 is not constant, but varies with thevalue of X3. To avoid overcomplicating the discussion,we consider only its value in the neighborhood of X30. This is

- aby3/[(y3 - b) sin2w - 2by3 cos2w].

It is obviously different from the expression obtainedfor the other plane in Eq. (19). In fact, the two areequal only for the two points for which y3 = b cotw,or y = b sinw/(1 - cosw). Both these values ofy3 are positive, so neither point falls in the regionactually observed by the user of the microscope.

In any practical case, the differences in magnificationare probably unnoticeable.

When X3 = 0, the expression for y4 becomes identicalwith that in Eq. (17), as is to be expected. Conse-

May 1969 / Vol. 8, No. 5 / APPLIED OPTICS 917

Page 6: Stereoscopic Imagery in a Type of Stereoscopic Microscope

quently, the expression for the longitudinal magnifica-tion given in Eq. (20) holds here also.

Without writing down the expression for the ratioof longitudinal to lateral magnification, it is evidentthat it must differ from that given for the other plane inEq. (21).

Enhancement of Depth PerceptionThe effectiveness of the instrument in aiding the

user to detect small differences in depth is unchangedby making the eyepiece axes converge, for the changein angle of convergence of the lines of sight in lookingfrom one point to another remains unaltered.

Summary

When the eyepiece axes are parallel to each other,Eqs. (9) gives the transformation from object spaceto the space of the stereomodel. The transformation isnot linear, so the model is not simply a scaled-up three-dimensional image of the object. Lines in the objectspace parallel to the axes of the magnifier are trans-formed into lines which, extended, pass through afinite point in the model space. Lateral magnificationvaries inversely with the distance of the object planefrom the first principal focus of the magnifier. Longi-tudinal magnification is proportional to the square ofthe lateral, but, unlike the case of a single lens, theconstant of proportionality is not generally unity.The ratio of longitudinal to lateral magnification isinfinite when the object point is at the first focal pointof the magnifier, and drops to unity at a distance from

this point which depends on the monocular magnifyingpower of the microscope.

The effectiveness of the instrument in enhancing theability to detect small differences in depth is propor-tional to the monocular magnifying power of the instru-ment and to the angle of convergence of the two view-ing axes in the object space.

When the eyepiece axes are made to converge, thedefinition of the stereomodel as given breaks downexcept in two planes, one being the plane of symmetryof the instrument, and the other, containing the eye-piece axes, being perpendicular to it. In the plane ofsymmetry, imagery is qualitatively like that in thecase of parallel eyepiece axes. However, it is possibleto choose the parameters of the instrument so thatthe point of equal longitudinal and lateral magnifica-tion falls within the depth of field of the instrument. Inthe plane containing the eyepiece axes, lines which werestraight in the object are imaged as curves. Also, thelateral magnification for a given value of y differsfrom that in the plane of symmetry. When the angleof convergence is small, these anomalies will also besmall.

References

1. A. W. Judge, Stereoscopic Photography (Chapman and HallLtd., London, 1950), Chap. 2.

2. 0. von Gruber, Photogrammetry (American PhotographicPublishing Company, Boston, 1949), Chap. 10. This dis-cussion is hard to follow in places, perhaps because of choiceof language in translation from the German.

SYMPOSIUM ON QUANTUM OPTICS

to be held as part of the APS meeting in Rochester, N.Y.,18-20 June 1969

E. T. JAYNES (Washington University)

The Lamb Shift in Semiclassical Theory

L. MANDEL (University of Rochester)

B. B. SNAVELY (Eastman Kodak Company)

Stimulated Emission from Organic Molecules

E. L. HAHN (University of California at Berkeley)

Self-induced Transparency

Optical Photon Correlation Measurements

G. B. BENEDEK (Massachusetts Institute ofTechnology)

On the Spectrum of Light Scattered from aStationary Fluid, a Turbulent Fluid, and from aFluid Interface

A. J.> DeMARIA (United Aircraft Corporation)

Progress on Picosecond Laser Pulses

E. WOLF and G. S. AGARWAL (University ofRochester)

Phase-Space Methods in Quantum Optics

918 APPLIED OPTICS / Vol. 8, No. 5 / May 1969