Evaluation of Hologram Imaging by Ray Tracing 1. A. Abramowitz This paper applies the methods of ray tracing to the evaluation of aberrations produced by Fresnel holo- grams that are recorded on two dimensional media. Attention is given both to holographic systems that employ lenses to modify the object wave and to lensless systems. Two lensless systems are discussed. The first, a system operating at optical wavelengths, is of numerical aperture 0.025, magnification 25, and unity wavelength ratio, X recording/X reconstructing. The field size of this device is better than 300 times the resolution; the system is diffraction-limited for points within a depth of 2.5 cm about the nominal object point. The second system, designed to record at x-ray wavelengths and to reconstruct at optical wavelengths, is of numerical aperture 0.0125, magnification 1000, and wavelength ratio 0.001. The field size is better than 3000 times the resolution; the system is diffraction-limited for points within a depth of 0.035 cm about the nominal object point. The performance of a lensless system is compared to that of a holographic system which employs an analytic aplanatic lens of high numerical aperture to form an enlarged real image of the object which is then recorded by a holographic system of unity magnification. The distortions of such a system are those of the lens alone. A method is established to design and ray trace an aplanatic lens. The system with such a lens is found to be in every way superior, at optical wavelengths, to the comparable lensless system. 1. Introduction Since the invention of holography, its application to the field of microscopy has been often discussed. -' A holographic microscope has the advantage of recording information about all three dimensions of an object. Furthermore, a holographic device may work outside the range of optical wavelengths. As in other optical sys- tems, geometrical distortions may be expected to limit the performance of a holographic microscope of high resolution. For resolution of the order of a wavelength, a holo- gram of size comparable to the object-to-hologram dis- tance is required. Previous analyses of geometric dis- tortions of holograms by Meier," Champagne, 9 and Leith et al. 4 are based upon the Taylor series expansion, which cannot be applied if the distance from the object to the center of the hologram is less than one-half the height of the hologram. Other workers have con- sidered the alternate possibility of describing the geo- metric distortions of holograms by ray tracing. Hel- strom1 0 has presented the requisite set of equations and Offner" has described the procedure for analyzing the virtual image of a hologram by means of a ray trace. He investigated the distortions of the virtual image The author was with the Electrical Engineering Department, Cornell University, Ithaca, New York when this work was done; he is now with Hughes Aircraft Company, Aerospace Group, Culver City, California 90230. Received 6 September 1968. formed by a particular hologram as a function of the reproducing laser. The real image can be analyzed by the same method if the negative of the diffracting power, 1 as evaluated from his Eqs. (9) through (14), is used in Eqs. (15) and (16).1 A suitable translation of axes yields the Helstrom results. This paper extends the methods of ray tracing to a point at which Fresnel holographic systems of specific design characteristics may be optimized. The position of the image plane yielding minimum geometric distor- tions, and the magnitude of the distortions in this optimum plane are calculated as a function of geometry and wavelength ratio for systems of specified magnifica- tion and numerical aperture. From these data the per- formance and geometry of the optimum system of particular numerical aperture and magnification are determined. Lensless holographic microscopes have been con- structed in the laboratory. Leith and Upatnieks 3 have successfully resolved a bar grating to the order of 3 ,, but this work has ended without further contribu- tions to the design of a practicable system.' 2 This paper demonstrates the difficulty in forming a magnified real image of high resolution by lensless holography. Fortunately, holographic configurations of unity mag- nification have been shown to produce no geometric dis- tortions of their own. 48 Both the groups of Carter and Dougal 7 and of van Ligten and Osterberg 5 ' 6 (the latter have a system whose resolution is better than 1 ,) have mounted objects in the stage of a conventional microscope; the magnified image was employed as the February 1969/ Vol. 8, No. 2 / APPLIED OPTICS 403
Transcript
Evaluation of Hologram Imaging by Ray Tracing
1. A. Abramowitz
This paper applies the methods of ray tracing to the evaluation of aberrations produced by Fresnel holo-grams that are recorded on two dimensional media. Attention is given both to holographic systems thatemploy lenses to modify the object wave and to lensless systems. Two lensless systems are discussed.The first, a system operating at optical wavelengths, is of numerical aperture 0.025, magnification 25, andunity wavelength ratio, X recording/X reconstructing. The field size of this device is better than 300 timesthe resolution; the system is diffraction-limited for points within a depth of 2.5 cm about the nominalobject point. The second system, designed to record at x-ray wavelengths and to reconstruct at opticalwavelengths, is of numerical aperture 0.0125, magnification 1000, and wavelength ratio 0.001. The fieldsize is better than 3000 times the resolution; the system is diffraction-limited for points within a depthof 0.035 cm about the nominal object point. The performance of a lensless system is compared to thatof a holographic system which employs an analytic aplanatic lens of high numerical aperture to form anenlarged real image of the object which is then recorded by a holographic system of unity magnification.The distortions of such a system are those of the lens alone. A method is established to design and raytrace an aplanatic lens. The system with such a lens is found to be in every way superior, at opticalwavelengths, to the comparable lensless system.
1. Introduction
Since the invention of holography, its application tothe field of microscopy has been often discussed. -' Aholographic microscope has the advantage of recordinginformation about all three dimensions of an object.Furthermore, a holographic device may work outside therange of optical wavelengths. As in other optical sys-tems, geometrical distortions may be expected to limitthe performance of a holographic microscope of highresolution.
For resolution of the order of a wavelength, a holo-gram of size comparable to the object-to-hologram dis-tance is required. Previous analyses of geometric dis-tortions of holograms by Meier," Champagne, 9 andLeith et al.4 are based upon the Taylor series expansion,which cannot be applied if the distance from the objectto the center of the hologram is less than one-half theheight of the hologram. Other workers have con-sidered the alternate possibility of describing the geo-metric distortions of holograms by ray tracing. Hel-strom10 has presented the requisite set of equationsand Offner" has described the procedure for analyzingthe virtual image of a hologram by means of a ray trace.He investigated the distortions of the virtual image
The author was with the Electrical Engineering Department,Cornell University, Ithaca, New York when this work was done;he is now with Hughes Aircraft Company, Aerospace Group,Culver City, California 90230.
Received 6 September 1968.
formed by a particular hologram as a function of thereproducing laser. The real image can be analyzed bythe same method if the negative of the diffractingpower, 1 as evaluated from his Eqs. (9) through (14), isused in Eqs. (15) and (16).1 A suitable translation ofaxes yields the Helstrom results.
This paper extends the methods of ray tracing to apoint at which Fresnel holographic systems of specificdesign characteristics may be optimized. The positionof the image plane yielding minimum geometric distor-tions, and the magnitude of the distortions in thisoptimum plane are calculated as a function of geometryand wavelength ratio for systems of specified magnifica-tion and numerical aperture. From these data the per-formance and geometry of the optimum system ofparticular numerical aperture and magnification aredetermined.
Lensless holographic microscopes have been con-structed in the laboratory. Leith and Upatnieks3
have successfully resolved a bar grating to the order of3 ,, but this work has ended without further contribu-tions to the design of a practicable system.' 2 Thispaper demonstrates the difficulty in forming a magnifiedreal image of high resolution by lensless holography.Fortunately, holographic configurations of unity mag-nification have been shown to produce no geometric dis-tortions of their own.4 8 Both the groups of Carterand Dougal7 and of van Ligten and Osterberg5'6 (thelatter have a system whose resolution is better than 1 ,)have mounted objects in the stage of a conventionalmicroscope; the magnified image was employed as the
Fig. 1. (a) The location of the physical elements used in therecording of the hologram. The coordinates in the plane of thehologram are (x,y,0). Laser Xi is located at the point (-u,-v,-p) and the object is located at the point (-xo,-yo,-q).XP(xy) and X Q(x,y) are measured, respectively, from the laserXi and from the object to the point (x,y,O) on the hologram. POand Qo are the distances from the laser X and object to the centerof the hologram. (b) The location of the physical elements usedto form a real image from the hologram. Laser 2 is located atthe point (-u',-v',-p'). 2 P'(x,Y) is the distance from thelaser X2 to the point (x,y,O) on the hologram, and P' =X2P'(d/2,0). 2 Q'(x,y) is the length from the point (x,y,0) on
the hologram to the point (x',y',q') in the image space.
object of a holographic system of essentially unitymagnification. In such cases, the lens' distortions alonewould limit the over-all performance of the system.
In order to investigate the distortions of such a systemand to compare its properties with those of a lenslesssystem of similar design parameters, without the com-plexity of ray tracing multielement objectives, theauthor defines an analytic lens as one which, at itscorrected conjugates, exhibits no spherical aberrationand which satisfies the Abbe sine condition. The sur-faces of such an aplanatic lens may be computed bythe method of Wasserman and Wolf.3' 14
II. Theory
A general Fresnel holographic system is described byFig. 1. 0, the object, is assumed to be a point locatedat (-x0 , - yo, - q), which scatters the incident radiation
as a spherical wave, and the lasers are assumed to bepoint sources which emit spherical waves. The wave-length (X2) of the reconstructing laser, located at(-u', -v', -p'), differs from that () of the referencelaser which is located at (-u, -v, -p).
The total optical length of any ray in the real imagespace of a holographic system, from which the positionof the real geometric image may be determined, is nowevaluated.
The optical length P(x,y) from the laser Xi to thepoint (x,y,0) on the hologram is
P(X,y) = (l/Xi)[p2 + ( + U)' + ( + V)2].
Optical length Q(x,y) from the object point to the point(x,Y,O) is
Q(X,y) = (/X0)[q2 + (x + Xo)I + ( + y)2]l.
The optical length P'(x,y) from the laser X2 to the point(X,y,0) is
P'(X,y) = (1/X2)[p'2 + (x + u) 2 + (y + v')2]1.
The optical length Q'(x,y) from the point (x,y,0) of thehologram to a point (x',y',q') in the real image spaceis
Q'(x,y) = (1/X2)[q2 + (x - x') 2 + (y - y) 2] 1.
Throughout this work the x-z plane will be maintainedas a plane of mirror symmetry.
From the work of Champagne,9 the lateral magnifica-tion M between planes which are parallel to the holo-gram is
M [1 Q(d/2,0) F 2 Q(d/2,0) - (1)P(d/2,0) X2
2P'(d/2,0)j
X2 2
Q'(d/2,Q) (2)
X12 Q(d/2,0)
The upper sign corresponds to one of the first order dif-fraction images and the lower sign to the other. Byinspection of Eq. (2), a real image [Q'(d/2,0) > 01formed by the first diffraction term requires positivemagnification. To form a real image from the secondterm requires negative magnification. For the realimage corresponding to the interference term of theupper sign.
M= -Q ()
M 1i- ( _ X12 2X2 2 , (do'), (3a)
M> 0;
and for the real image corresponding to the interferenceterm of the lower sign:
Fig. 2. The location of the physical elements of a holographicmicroscope with an objective lens; (a) recording, (b) reconstruc-
tion.
If, in order to form diverging beams (which are neces-sary to illuminate large holograms), both p and p'are held positive, of the two possible geometries that ofEq. (3a) has the minimum sum Q[(d/2),O] { [1/P(d/2,0)] + (X'/X 2
2) [1/P'(d/2,0) ] } for a specifiedlateral magnification. The illuminating lasers may beplaced further from the hologram center, which in turnpermits the largest possible hologram aperture. Inaddition, only Eq. (3a) includes the two cases of perfectimaging of the real image (as discussed by Leith et al.4and by Abramowitz and Ballantyne") which provides,at least, a promising starting point. Consequently,the representative lateral magnification M of a lenslesshologram will be determined by Eq. (3a).
The ray tracing equations as applied to the problemof optimizing the real image of a holographic micro-scope have already been discussed. 6 In the presentwork as previously, the position of the image plane wasoptimized by maximizing the value of a quality functionF, defined in the image space by5 16
100
F = E (r, -r ave + Ar)'. (4)100 =i
In Eq. (4), r is the vector from an arbitrary origin Nin the image plane to the intersection with this plane ofthe ray from the center of the th hologram section;r ave is the vector from N to the center of gravity ofthe r, and Ar is an arbitrary constant which limits themaximum value of the quality function. A systemwhich is optimized by the quality function F imagesa point object into a spot whose dense central core issmaller than that of a system which is optimized by thestandard deviation.' 6 Consequently, a microscopewould be evaluated by the function F because the imagesof two adjacent object points of the former system may
be more easily resolved than the two of the latter. 6
F- is approximately the radius of the central core (thegeometric spot size) of the geometric image of a pointobject.
For a specified geometry, it is always possible to im-prove performance by scaling down the elements of thesystem. The ray tracing equations are functions onlyof ratios of the system's dimensions and of the wave-length ratio. At fixed X1/X2, halving all physical dimen-sions halves the spot size. The ultimate limits on thisprocess are set by physical realizability and film res-olution.
A. The Analytic Aplanatic Lens
Holographic systems may employ objective lenseswhich provide part or all of the image magnification.The real (or virtual) image of the lens becomes theobject of the hologram (cf. Fig. 2). A hologram of unitymagnification produces no distortions of its real image ifthe reference and reconstructing waves are each plane,their respective wave vectors lie in the x-z plane andare at equal but opposite angles to the z axis, and thewavelength ratio is unity. The case of cosa = - cos#= 0 is excluded because then the distortionless imagewould fall in the path of the transmitted reconstructingbeam and the virtual image. Thus, for the cases ofperfect geometric holographic imaging, if the objectivelens provides all of the magnification, those geometricdistortions present are due only to the lens. It isdesirable to calculate by ray tracing the geometric dis-tortions in the image plane of a lens so that the over-allperformance of such a system could be compared with alensless design of similar parameters. This sectiondescribes a simplified procedure for the ray tracing ofobjective lenses.
The design of a lens is indeed complex. Could alllenses of a particular focal length, numerical aperture,and magnification (which determines the conjugate foci)be characterized and ray traced without necessarily in-vestigating specific designs? Any good objective lensmust, first, be corrected for spherical aberration and,second, for adherence to the optical sine condition(which assures that points near the optical axis have aconstant magnification). No objective may be cor-rected at more than one pair of conjugate points, butonly two surfaces are necessary to correct for bothspherical aberration and coma (deviation from the sinecondition) at a designated pair of conjugates. "4 Sincelaser applications do not require chromatic correction,the most simple analytic model employs a singleaplanatic lens to provide the necessary geometric cor-rections. The method of Wasserman and Wolf Il14 isemployed to calculate the requisite surfaces.
The distances q and q' to the conjugate foci (theimage is chosen to be real) satisfy the lens equation 1/q± l/q' = 1/f and are chosen such that the ratio q'/q
is equal to the corrected magnification M. The axialseparation Az between the correcting surfaces is chosenby trial and error to be at least large enough to diffractall rays that enter the aperture.
Any number of points on and tangents to the entranceand exit surfaces of the aplanatic lens may be calculatedand tabulated by the method of Wasserman andWolf. "4 The ray from any axial object point to oneof the points on the entrance surface may be easilytraced through the first surface, but the ray would notnecessarily intersect the exit surface at one of the com-puted points. Consequently, the procedure chosen wasto fit, by the method of least squares, a polynomial ofthe form
z= A m(y,")' n • 20m=0
to the calculated points (n',zn') on the exit surface.Rays may then be easily traced through the lens.
B. The Required Position of the Reference Laser
The means by which the real image is separated fromthe virtual and zero order images lies exclusively withinthe first stage of the holographic process. When thereference wave is plane, Leith and Upatnieks havedescribed the interference between the object wave andreference wave as a modulation of a spatial carrier (thereference wave) by the spatial frequency spectrum ofthe object.17. 8 This is also a valid formulation of theimage formation in a system in which a diverging refer-ence wave is employed.
The object transmittance is specified by a superposi-tion of plane waves; the spatial frequency energydensity spectrum of the object wave is unchanged withz' (Ref. 19). Therefore, the spatial frequency spectrumT'(e,,q) of the object wave at the hologram plane isequal to the product of T(e,-q), the spatial frequencyspectrum of the object, and an exponential phase factorexp [j0(e,-q)]: T'(E,fl) = T(e,rq) exp [(e,,7)]. The spatialfrequency bandwidth of the wave remains that W ofthe object. The linear interference terms beween theobject and reference waves are proportional to
respectively. The factor exp [-j27rP(x,y)] may bebrought within the integral, since it is not a function ofe and -q. If in the incremental region (x,by) about(x,y,0), P(x + x, y + by) is approximated by
P(X,y) + bP(xsv ax + bP(x,y b)y
the spatial frequency spectrum of the first linear inter-ference term at (x,y,0) may be described as a modula-tion of the reference wave by the object spectrum suchthat the object spectrum is shifted in spatial frequency
space by an amount designated as the two dimensionalspatial carrier frequency (f.,fJ):
(fife) = P(), P(XY
The spatial frequency spectrum of the second linearinterference term is the negative of the first.
Each of these spectra corresponds to a first order(linear in the spatial carrier frequency) image. Inaddition, the zero order Gabor images, which are derivedfrom the nonlinear term
j f Z ' f ::xi T(e,,7) exp[j(e7)l exp[j27r(xE + y)ldEd?7
of the square law detector film, are centered about (0,0)in spatial frequency space with a bandwidth of 2Wcaused by intermodulation products. 1 To separate thereal first order image spectrum from the spectra of thevirtual and zero order images produced by the differ-ential hologram at (x,y,0), it is necessary to meet theinequality:
[1P(xy)/1x] = f > 3W/2,
or[bP(xy)/x] = f > 3W/2.
f., (or f) is 1/Xi times the directional cosine with thex (or y) axis of the ray from the reference laser to thepoint.
Although the object bandwidth may be of the order of2/X, the actual usable object bandwidth is only twicethe inverse of the resolution of the optical system. Anyobject bandwidth in excess of this amount will not bepassed through the aperture, whose action may belikened to that of a linear filter of spatial frequencies.20The first and zero order image spectra may be over-lapped by an amount equal to this excess without de-grading the image quality.
At the infinitesimal hologram about the point (x,y,0)the reconstructing wave is modulated by the spatialfrequency spectra of interference terms of all orders.Each spectrum is shifted in spatial frequency space byan amount (f.',fJ') equal to
(f-xv")= P'(XY), PI(xy)].
Unfortunately, the spatial carrier frequencies at differ-ent points (x,y,0) are unequal if the reconstructingwave is not plane. The disastrous overlap between thezero and first order spectra of different points may beavoided if the spacing between the spectra in the firststage of the holographic process is made larger than theabsolute value D of the difference between the largestand smallest spatial carrier frequency f' (or f,'): D =(f') max - (fx') minl.
Therefore, the first order real image of the entirehologram may be removed from the noise of the virtualand zero order images if the directional cosines with thex (or y) axis of the rays from the reference laser to allpoints (x, y, 0) on the hologram are greater than
Fig. 3. (a) The location of the physical elementslensless holograms by Systems 1 and 2. (b) Thephysical elements used to reconstruct the hologra1 and 2. System 1: a holographic microscopeaperture 0.025, magnification 25, and unity wa'u = 11.00 cm, xo = 10.56 cm, p = 61.24 cm, q =
= 0.2, q' = 1476 cm, x' = -301.2 cm, and d = 3ca holographic microscope of numerical aperturefication 1000, and wavelength ratio of 0.001. uxo = 5.25074 cm, p = 59.7590 cm, q = 59.6992 c
V = 59.5763 cm, x' = -5.98788 cm, and d =
C. Specific Lensless Holographic Sysi
In this section, the details of the design trative and potentially useful lensless holetems are described. Use is made of all conviously derived. The systems to be discuss
(1) A system of small numerical apertumagnification, unity wavelength ratio, and Ifield. This is, of all systems, the most easilthe laboratory.
(2) A system of high magnification (derfrom the- wavelength ratio), low numericand moderate depth of field. Such a possibly record with x rays (5 X 10-1construct with light (-5 X 10-5 cm).
(3) A system of low magnification, butnumerical aperture.
Assume that the object extends dowiwhat is designated as the object point and isby some means other than by the reference back end of the reference laser). Then,avoid interference between the reference willuminating beam at the object and to illentire film surface, it is necessary that (cf.> u/p. The greatest number of fringes
HOLOGRAM when (xo/q) is set equal to (u/p) (this permits the largecentral fringes to be recorded).
The computer analysis proceeds as follows. First,the object distance XI Q(d/2,0), the size of the hologramd and hence the numerical aperture, and the magnifica-tion M were fixed. For each chosen value of X1/X inthe range 0 < XI/X2 < 1 (which permits the wavelengthratio to contribute to the magnification), the values ofP(d/2,0) and P'(d/2,0) were altered under the con-straints of Eq. (3a). To further optimize the geometricconfiguration of the system, the reference and recon-structing lasers were moved one at a time in the x-zplane along circles of radii XP(d/2,0) and X2P'(d/2,0)respectively.
For each configuration, the image plane is optimized.The image spot must fall with its center of gravity onthe mirror plane x-z. First, the intersection of eachof the 100 image rays with this plane is calculated. The
xi intersection nearest in z and that furthest in z from theIMAGE POINT hologram are next chosen as limits to the range in which
the quality function is evaluated at equally spacedimage planes.
The distances XQ(d/2,0), XP(d/2,0), and X2P'(d/2,
used to record 0) of the object, reference laser, and reconstructinglocation of the laser, respectively, from the center of the hologram arems of Systems henceforth to be represented by the terms Qo, Po, and
of numerical Fot.velength ratio.58.78 cm, cos-Y 1. System 1m. System 2: Figure 3 sets forth the positions of the elements of the0.0125, magni- holographic system of numerical aperture 0.025, mag-= 5.25601 cm, nification 25, and unity wavelength ratio which wasm cosY = 0.1, found by ray tracing to be optimum. The resolution= 1.5 cm. is 20 XI (Fraunhofer diffraction-spot halfwidth of a
rectangular aperture) which at optical frequencies isapproximately 10-' cm. Hence, the size of the diffrac-tion spot (the product of the resolution and the mag-
tems nification) is of the order of 25 X 10-' cm. A diffrac-If three illus- tion-limited system would require the geometric spot,graphic sys- radius [-F-' as defined by Eq. (4) ] to be less than that.straints pre- of the diffraction spot. In this system, F'1 < 25 Xed are: 10-' cm. Therefore, F > 40 cm-'.re, moderate Table I presents the values of the quality functionhigh depth of for points directly below the object point. Interpola-ly realized in tion between the values of this table indicates a field size
ived entirely Table I. A Comparison, for System 1, of the Image Positionsal aperture, (x'), Quality Functions (F), and Magnifications (M) of Points in
col the Object Field Which Are a Distance X Below the ObjectI c Point,
cm) and re-
of very high
award from. illuminatedvave (by thein order toave and theiuminate theFig. 1): xo/qis recorded
T X/ F(cm) (cm) (cm-) M
0 -301.22499 2.639 X 10210-5 -301.22525 2.643 X 10' 2510-4 -301.22750 2.679 X 102 25.110-3 -301.25009 2.887 X 10' 25.110-2 -301.47598 1.790 X 10' 25.110- -303.73481 6.187 X 101 25.1100 -326.32324 7.056 25.1
a The hologram and image1475.67798 cm.
plane are parallel and separated by
February 1969 / Vol. 8, N. 2 / APPLIED OPTICS 407
_ D
Table II. A Comparison, for System 1, of the Quality Functionsof Points in the Object Field Which Are to the Back and Front of
the Object Pointa
q q' Quality function(cm) (cm) (cm-,)
61.00 1.178 X 104 5.18260.50 4.666 X 103 1.688 X 10'60.00 2.876 X 103 5.565 X 10159.50 2.066 X 103 1.502 X 10259.00 1.627 X 103 2.125 X 10258.00 1.115 X 103 1.155 X 10257.00 8.425 X 102 7.277 X 10'56.00 6.726 X 102 7.468 X 101
a q is the distance from the point to the plane of the hologram.The hologram and image plane are parallel and separated by thedistance q'.
of the order of 0.36 cm, which is approximately 360times the resolution.
Table II presents the values of the quality functionfor points that are to the back and front of the objectpoint. Each point is imaged on an optimum plane andeach set of conjugate planes has a different numericalaperture and magnification, which requires a modifica-tion of the inequality between the quality function Fand the radius of the diffraction spot. In the longi-tudinal direction, the system remains diffraction-limitedover a range of 5 cm.
A reduction of the value of u would greatly improvethe geometric distortions of the system, because if thevalue of u were reduced to 6.38 cm, the hologram couldbe increased in size from 3 cm to 6 cm square and thesystem would be diffraction-limited at the new aperture.u, however, is constrained by Eq. (5) to separate thereal image from noise. It may be possible to toleratesome interference from the first order virtual and thetwo zero order images; this might best be determinedfrom experiment. At any rate, the improvement in theresolution is considerably less than an order of magni-tude, although this might still be a significant improve-ment.
2. System 2
This system is designed to make use of a large wave-length ratio X2/Xl to provide the necessary magnifica-tion. In addition, good resolution which is a functionof the smaller wavelength Xi may be obtained. Ideally,the system would record the hologram at x-ray fre-quencies (if sufficient coherence exists) and would re-construct at optical frequencies.
Figure 3 is a schematic representation of the systemof numerical aperture 0.0125, magnification 1000, andX,/X2 of 0.001 which was found by ray tracing to beoptimum. The resolution of this device is 40 Xi, whichat x-ray frequencies is approximately 2 X 10-6 cm.The corresponding diffraction spot is 2 X 10-3 cm inradius; F, the quality function for this system, must begreater than 500 cm-'. Tables III and IV indicate,respectively, a field size of approximately 0.77 X10-2 cm (more than 3800 times the resolution of the
system) and a longitudinal range of diffraction-limitedperformance of 0.07 cm.
Systems 1 and 2 demonstrate that, at small apertureand at moderate to high magnification, the configura-tion which gives the best image quality requires planewave reconstruction. This result may be explained asfollows: the reference and object waves are essentiallyconcentric spherical waves (cf. Fig. 3), resulting in alinear interference term that, at the recording film,approximates a plane wave. The slight variation inphase of this wave over the hologram resembles thatof the field from an object point which is at a large rela-tive distance from the hologram and which is offset fromthe z axis by an angle comparable with that of the ob-ject offset. In these systems, the reconstructing planewave is offset from the z axis at an angle which is nearlyequal and opposite to that of the object. Conse-quently, the real image (if reconstructed at unity wave-length ratio) is at a large distance from the hologramand is relatively close to the z axis. If reconstructiontakes place at a larger wavelength, the diffraction bythe hologram fringes becomes more effective and theimage point lies closer to the hologram. As the aper-ture is increased in size or as the reference laser is dis-placed from the z axis, the approximation of concentricspherical waves becomes less valid over the extent of
Table IlIl. A Comparison, for System 2, of the Image Posi-tions (x), Quality Functions (F), and Magnifications (M) of
Points in the Object Field Which Are a Distance rBelow the Object Pointa
T xi F(cm) (cm) (cm-') M
0 -5.98788 3.239 X 10310-' -5.98798 3.238 X 10' 100010-6 -5.98888 3.236 X 10' 100010-5 -5.99790 3.208 X 103 100210-4 -6.08810 3.126 X 103 100210-' -6.99277 3.027 X 10' 100510-2 -16.4499 7.003 X 101 1046
a The hologram and image plane are parallel and separated by59.57625 cm.
Table IV. A Comparison, for System 2, of the Quality Functionsof Points in the Object Field Which Are to the Back and Front
of the Object Pointa
q q' Quality function(cm) (cm) (cm-)
59.7442 2.357 X 102 1.200 X 10259.7342 1.426 X 102 3.518 X 10259.7242 1.022 X 102 1.242 X 103
59.7142 7.954 X 101 3.282 X 10'59.7042 6.502 X 101 2.682 X 10359.6942 5.499 X 101 3.554 X 103
59.6842 4.759 X 101 2.272 X 10359.6742 4.190 X 101 8.057 X 102
a q is the distance from the point to the plane of the hologram.The hologram and image plane are parallel and separated by thedistance q'.
the film, and distortions increase. Eventually, otherconfigurations are required to optimize the system.
As in the case of System 1, a decrease in the value ofu in System 2 leads to a reduction of the geometric dis-tortions at larger numerical apertures, but again,resolution is not significantly increased because theapparent concentricity of the object and reference wavescan be little improved.
3. System 3
For best resolution, the object position x0 must belocated within the range 0 > xo - d (near the centerof the recording film), so that the hologram subtends aslarge a solid angle about the object as is possible. Ifthe reference laser is placed at a position that satisfiesEqs. (3a) and (5), a lensless system of large numericalaperture might in principle be developed. This wasnot however possible, as explained below.
During a first set of computations in which thewavelength ratio X1 /X2 was varied, a particular difficultywas encountered: there was no diffraction from specificportions of the hologram. An alternate statementof the geometric theory of holography treats the holo-gram as a modulated diffraction grating:" the rela-tionship among the period along x, k, of the infinitesimalgrating at the point (x,y,0); the reconstructing wave-length, X2; the directional cosine to the x axis 0 of theray to the hologram from laser X2; and the directionalcosine to the x axis, 01 of the nth order diffracted rayis" 0,, = n,X2/k + i. Similarly, to the y axis,"02, = n2/k. + 02; subject to the condition,
Oia2 + 2n' < 1. (6)
If the condition in Eq. (6) is violated, no diffraction(in geometric theory) takes place and an evanescentwave is produced. Since X1/X2 < 1, it is quite possiblefor the interference between the object and referencewaves to produce fringes that do not diffract duringreconstruction (if either k,, or k, is less than X2/2, norays diffract).
This phenomenon restricts the possible geometricconfigurations of a holographic system, which removesmuch of the freedom permitted by the wavelength ratio;distortions tend to decrease with a reduction of the ratioX,/X2, but a value is soon reached at which the gratingperiod becomes prohibitively small for diffraction. Fora system of numerical aperture 0.67, magnification 4,and X/X2 of 0.3043, the geometric distortions were, atbest, 5000 times greater than the diffraction spot radius.The holographic system of unity wavelength ratio isless restricted by Eq. (6). The ratios 2/k,, and 2/kydecrease when Al/X2 is increased to unity, which, in turn,permits larger apertures and more varied geometricconfigurations. Image quality, however, is onlyslightly improved (distortions are greater, by a factorof 3500, than the diffraction spot).
D. Ray Traces Through an AnalyticAplanatic Lens
If the magnified real image of an object is employed asthe object of a holographic system of unity magnifica-
Table V. A Comparison, for an Analytic Lens of NumericalAperture 0.5, Focal Length 0.8 cm, and Magnification 21 X
of the Quality Functions of Axial Points in the Object SpaceWhich Are to the Back and Front of the Corrected Pointa
q q, Q' Quality function(cm) (cm) (cm) (cm-')
1.0381 3.488 3.087 1.353 X 104
0.9381 5.434 4.440 1.246 X 104
0.9131 6.459 5.647 9.319 X 104
0.8881 8.065 7.462 7.445 X 104
0.8781 8.995 8.517 4.487 X 1050.8681 10.20 9.842 4.933 X 10'0.8581 11.82 11.60 9.187 X 104
0.8481 14.11 14.03 2.929 X 1050.8431 15.65 15.63 2.060 X 100.8381 17.60 17.60 9.482 X 100.8331 20.14 20.08 1.968 )< 100.8281 23.58 23.37 1.229 X 1050.8181 36.17 36.13 3.369 )< 1030.8131 49.67 44.57 7.711 X 1030.8081 79.86 89.53 2.062 X 102
a q is the distance from the axial point to the entrance of thelens (z = 0); q' is the distance from the exit of the lens (z' = 0)to the nominal image point (/q' = /f - /q); Q' is the dis-tance from the exit surface of the lens to the point of best imagequality.
tion, the geometric distortions are those of the objec-tive lens alone. To compare conveniently the perfor-mance of such a system with that of a lensless design ofsimilar parameters, it is desired to ray trace such anobjective without investigating the particular designof its components. The results of the ray trace of ananalytic aplanatic lens are presented in this section.
Following the method of Wasserman and Wolf,3' 1 4
the surfaces of an analytic lens of numerical aperture0.5, focal length 0.8 cm, and magnification 21 X werecalculated. The index of refraction of the opticalmaterial was chosen as 1.5 and points on each surfacewere computed at increments of 0.0005 in the value ofthe sine of the angle between the traced ray, at theobject, and the optical axis.
An investigation of the results for a lens of thickness0.3871 cm revealed that the surfaces were smooth onlythrough the first 646 calculated points. The 647th andsucceeding rays could not be diffracted towards theoptical axis by the surfaces. Subsequently, the thick-ness of the lens A~z was increased to 300% of the diam-eter (z = 2.903 cm; cf. Fig. 2). Full aperture wasattained.
The points of the second surface were fit best by apolynomial of 16th order in ys", which fit the surface tobetter than 2 X 10-1 cm (1/2500 of an optical wave-length).
Values of the quality function for object points to theback and front of the corrected point are given in TableV; the system remains diffraction-limited over a longi-tudinal depth of 0.23 cm. The distance Q' from the exitof the lens (z' = 0) to the image point differs from thedistance q' to the nominal image point by no more than18.3%.
In summary, lensless holographic microscopes ofspecific numerical aperture and magnification wereoptimized by altering the wavelength ratio and the rela-tive geometry of the system under the constraints of themagnification and the necessity to s parate the realimage from noise. At each configurat on, the geometricdistortions of the system were determined by the eval-uation of a quality function whose value was usedto find the optimum position of the image plane.Next, the distortions of holographic systems whichemploy lenses to provide the entire image magnificationwere evaluated by ray tracing analytic aplanatic lenses.The surfaces of such a lens were first computed and thenray traced for object points along the optical axis. Againa quality function is used to determine the optimumposition of the image plane. The results of ray tracesfor representative systems of either type indicate thatat optical wavelengths an object of microscopic size canbest be imaged by the combination of a lens and ahologram of unity magnification rather than by a systemof lensless design. The former is the only feasiblemethod at high numerical apertures and it certainlymaintains its advantage at low numerical apertures.
The design of a lensless holographic system of highnumerical aperture was constrained by the necessityto separate the real image spectrum from the noise ofthe virtual and zero order spectra. Geometric dis-tortions, increased by the necessary offset of the refer-ence wave, prohibited diffraction-limited designs.
In the region of the electromagnetic spectrum inwhich lenses are ineffective as focusing elements,lensless holographic systems might be used with success.System 2, which is designed to span the extreme rangebetween x-ray and optical wavelengths, shows excellentfield size and good recordable diffraction-limited objectdepth.
Ray tracing retains its value as a tool to determinethe performance of holographic imaging systems of
specific characteristics. As with lenses, high speeddigital computers facilitate the design of optimum sys-tems.
The author wishes to thank Joseph M1. Ballantyne forhis encouragement, and the School of Electrical En-gineering and the Materials Science Center of CornellUniversity for their support.
Sections of this article were presented in a thesis tothe faculty of the Graduate School of Cornell Univer-sity for the Degree of Doctor of Philosophy in January1968.
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