wavefront investigation of a fourier transform lens with the fan trace interferometer

Wavefront Investigation of a Fourier Transform Lenswith the Fan Trace Interferometer

Karlheinz von Bieren

The performance of a Fourier transform lens is analyzed with the help of a new interferometric device

that records the aberration function along any selected section of the wavefront. These interferometricplots, which carry an automatic calibration, are analogous to the familiar geometric-optical fan ray tra-

ces and thus provide the means to compare the performance of completed lens systems with the design

criteria. The interferometer is particularly well suited to test Fourier transform lenses; however, it isreadily modified to investigate imaging systems as well. Therefore, photographic objectives or micro-

scopes may also be analyzed withalignment requirements.

I. Introduction

The interferometers for the investigation of wave-fronts can be grouped into two distinct categories.The first group utilizes the division of wavefrontprinciple. Young's historical experiment, which es-tablished the wave nature of light, belongs to thiscategory. Other representatives of this group areMichelson's stellar interferometer' and Rayleigh'sinterferometer for the precision gas index measure-ment.2

The second method is based on the division of am-plitude principle. Among the large variety of inter-ferometers that belong to this group there are twotypes that serve mainly to investigate lens aberra-tions: the Twyman-Green interferometer3 and theBates shear interferometer.4 The Twyman-Green, avariation of the Michelson interferometer, is used intesting optical systems in autocollimation. Thus,photographic objectives or telescopic systems may beexamined with it. However, due to the autocollima-tion feature, the exact interpretation of aberrations,as they appear in the interferograms, presents prob-lems.

This difficulty is overcome in the Bates shearinterferometer, which is a modification of the Mach-Zehnder type. In this instrument, a wavefront andits sheared replica interfere to produce straight frin-ges. The movement of this fringe pattern as a func-tion of the shear parameter makes it possible tocompute the wavefront slope. This is obviously a

The author is with Palisades Geophysical Institute, Sofar Sta-tion, Bermuda.

Received 27 December 1972.

this interferometer, which also provides considerable freedom from

tedious process, since the wavefront has to be mea-sured and computed for each point individually.Furthermore, the method is only applicable forlarge aperture systems, where there is sufficient roomto accommodate the interferometer between the lastrefracting or reflecting surface and the focal point ofthe wavefront.

The fan trace interferometer is based upon the di-vision of wavefront principle. Like the shear inter-ferometer, it measures the wavefront slope; however,the interpretation of the interferometer plot is ex-tremely simple. It requires no computation sincethe interferometer provides an automatic calibration.In addition, the alignment requirements for this in-strument are reduced to a minimum.

II. The Fan Trace Interferometer

The principle test method, which was used for theperformance analysis of a large aperture Fouriertransform lens, is shown in Fig. 1.

The Fourier transform lens L, which is to be test-ed, contains in its front focal plane (plane P1) anopaque signal film with two pin holes, which is illu-minated by coherent light of wavelength X and radi-an frequency w. The two pin holes are located orig-inally at the coordinates yl,zi and yl,zl + a.

The signal film is translating along the z-coordi-nate, and if we assume for mathematical conve-nience that the pin holes represent functions, theinput signal may be represented by

f(z, Y) = (Z - Z1 T)6(y - Y1)

+ (z - - a - r)b(y - y). (1)

The results in a previous paper5 indicated thatplane PI represents the entrance pupil for this sys-tem. Therefore, the exit pupil is at infinity. Ac-

1642 APPLIED OPTICS / Vol. 12, No. 7 / July 1973

COHERENTLIGHT _

ZE~~~~l A' ~~~ PLANE ,

PLANE P Q

f f

Fig. 1. Functional layout of the fan trace interferomneter.

cordingly, there exists an exact Fourier relation be-tween the exit pupil and the image plane. In thediffraction-theoretical analysis of lens systems, theexit pupil contains the aberration-free wavefront aswell as the aberration function. Thus, the imageplane contains the Fourier transform of these twocomponents. Since a Fourier relation exists also be-tween the entrance pupil and the image plane, theassumption that the entrance pupil contains the ab-erration function V(z,y) leads to identical results.Accordingly, the input function may be representedby

(zay) = [(z - z T)(y - Yi) + 6(z -z- a - r)(y - y)] exp[-j(27r/X)V(z 1y)]. (2)

Lens L performs a two-dimensional Fourier trans-form of this function, which leads to the followingexpression for the light amplitude in plane P2:

F(W~~~ aWY, z) = E expJ-j[w(z1 + b + T) + Wyy

b=O

+ (27r/X)V(z1 + b + y1)], (3)

where

WY = (27r/Xf)y'; w. = (27r/Xf)z'; b = 0,a.

A slit, which is located in the plane P2 , allows onlythe light at wy = iDy to expose the output film. Thisoutput film moves in synchronism with the inputfilm (either continuously or in discrete steps) alongthe y coordinate; therefore, the shift parameter appears as a coordinate on the output film and thetwo-dimensional intensity record on the output filmbecomes

r (z',T)= 2 + 2 cosl(27r/Af) z'a+ (2r/XV(z + T + a,yl) - V(z + ,yi). (4)

By overexposure as well as overdevelopment of theemulsion it is possible to record only the zero valuesof expression (4) or equivalently to record only thosepoints where the argument of the trigonometricfunction in Eq. (4) equals (2n + 1)7r, with n = 0, +1,+2 . . .

(271r/X)(a/f)z' + (2r/X)[V(z, + T +,a,y1 )- V(z + ,yi)] = (2n + 1)7r. (5)

Therefore, the following one-parametric family offunctions is recorded on the output film:

z (f/a)[(n + ) + V(z + + a,yl)

- V(z1 + T,yi) 1 . (6)

The geometrical layout in Fig. 1 is such that a fanat y = y is selected by the two scanning pin holes,and the interference pattern of the pinhole-emanat-ing rays is recorded at the coordinate wy = const. Itis obviously likewise possible to investigate thewavefront behavior in the perpendicular direction bya change in the orientation of the pinhole connectingline, output slit, and direction of output film travelby 900. In fact, the wavefront slope in any directionand along any track across the wavefront may be re-corded at any point in the output plane by a properchoice of the geometrical parameters of input andoutput.

The last term of Eq. (6) deserves special attention,because the expression

V(z1 + T + a,yl) - V(z1 + y1) (7)a

represents the slope of the line connecting the pointsof the wavefront, which were selected by the twopinholes. As the separation a is reduced to smallvalues, this expression approaches the differential

V(T)LIZ

Y=Yi

(8)

which represents the slope of the wavefront in theexit pupil as a function of r at the selected section y= yi. This is the expression that must be availableto transfer from a wave-theoretical treatment of thediffraction problem to the geometric-optical method.The lens design problem is generally solved by trac-ing rays through the optical system which penetratethe object point as well as selected points in the exitpupil and by plotting the intersection points of thoserays with the image surface. Therefore, the lens de-signer immediately starts with the geometric-opticalapproach ( - 0) to solve this diffraction problem,and the result is generally a spot diagram or a fantrace plot. Since the fan trace plot more readilyprovides relative phase information of the rays, it isthe preferred design tool if the aberrations are to beanalyzed or if the resulting diffraction picture mustbe computed. Since the light-propagating materialsinvolved generally are homogeneous within the re-fracting boundaries, the law of Malus may be ap-plied, which makes it possible to switch from a sys-tem of rays to their orthogonal trajectories, thewavefronts, or vice versa. This transfer can be per-formed at any point in the ray paths. For instance,if the treatment from object point to exit pupil wasof a wave-optical nature, the wavefront function xi'

July 1973 / Vol. 12, No. 7 / APPLIED OPTICS 1643

v.26mm v.52mm29 .0 Zg .0

.25 ou

-.250M

' 0Z,26MM(SIE COND ) ZI

Z,

5

-52_

_.25 015

52mm

- 250U

.25 OU

.

-.250O5

- 52m

-.25 OU

Fig. 2. Geometric-optical fan ray traces through Fourier transform lens.

= V(zl',yL') in the exit pupil (with coordinates z',yi') is available, and it is possible to compute the in-tersection points z',y' of any ray with an image sur-face at a distance p from the exit pupil with the helpof the equations

Z' - Z IV

p- x1' =~t y'- ' = -. (9)

p - xi' by,,

xl' in general is small in comparison with p, and itmay be neglected in Eq. (9) without introducing anappreciable error.

Z'= y1' - PG3V/5Y 1)' (10)

The wavefront function V contains the terms rep-resenting the Gaussian imagery as well as all theterms representing the aberrations of the optical sys-tem. If the ray convergence in the image plane isstigmatic for the selected object point, the wavefunction V must be such that the dependent vari-ables z' and y' in Eq. (10) remain constant (,y')for all values of z1' and yi'. If aberrations are preg-ent in V, a departure from these values 2' and y' re-sults. In general, each function in Eq. (10) dependsupon the two independent variables z1' and y 1'.

Z (Zi Y))y= (z1', Y1 ')(

Therefore, the lens designer normally designates oneindependent variable, e.g., Yi' as a parameter that iskept constant while the variable z' runs through itsrange of values. Thus, a set of fan ray traces resultswith a different set for each object point.


EO02,' 0

21 .25 ou

ZIE52_ .5m

-.25 u-.25 OU

.25 OU

Y1.52mZ'26mm(IE COND 2

Y'Er260M.26 _

MME Coo}

-52m

Yf26msZ1 *52am(SIE CO'D)

U--52m

-25 5

ZI

2; .25 CU

.25 OU

Zr-

-. 25s O

.2$OJ

ZE 52mm

-.25OU

YE-52 WAZ152mm(SWE Como) Zi

o - - -

-| -l -- l | w l_

l

i'

- ZE

t

f____1ZI

It is possible to select an image plane locationwhere the geometric-optical spread is smaller thanthe diffraction-optical half power spread of an ideallens. Thus, the wavefront slope remains within thelimits of X/4 per full aperture.

Fig. 3. Symmetrical Fourier transform lens.

The fan ray traces in Fig. 2 represent the aberra-tions of a symmetrical Fourier transform lens with afocal length of 1750 mm, a useful aperture of 100 mmX 100 mm in input and output plane and the generalgeometry of Fig. 3.

This lens was designed to operate in a matched fil-ter with a one-dimensional space bandwidth productof 2WT = 8000. The principal design considerationsfor this lens are described in Ref. 5. The large 2WTproduct makes it necessary to extend the operatingrange of the lens well beyond the limits of Gaussianoptics. This range is limited by a field angle a)which may be determined with the help of Eq. (12):

0 = (2X / D) 13 , (12)

where D indicates the aperture length in the inputplane.

Since matched filters are particularly sensitive tosmall errors in the spatial frequency locus, it is es-sential to recognize the fact that the imagery inplane P2 (Fig. 3) must contain a well defined amountof distortion. This precise amount of distortion isprovided automatically in a symmetrical design as inFig. 3. However, all too often this inherent distor-tion, which cannot be eliminated in a symmetricaldesign, is looked upon as a limit for symmetricalFourier transform. lenses; accordingly, most lensesoperate in a range where distortion is small enoughso that it can be neglected. In reality, however, dis-tortion provides the key for Fourier transform lensesto go beyond the limits of Gaussian optics. Thegeometric-optical fan ray traces through the lensof Fig. 3 reflect this property as well as a numberof third and fifth order aberrations for laser lightat 6328 A. Each ray fan in Fig. 2 emanates from anobject at infinity and penetrates the entrance pupilat a constant coordinate YE according to Fig. 4.

The incidence angles of the unrefracted rays havebeen selected such that the image coordinates ZI =0, z = 26 mm, and z, = 52 mm represent values im-posed by the sine condition requirement for the chiefrays. The calibration along the ordinates in Fig.2 is in optical units [OU], where 10U = X/(nsina.max)(n = index of refraction; amax = max. half-aperture angle). The diffraction-optical main lobespread covers a range of 1.0 along the ordinate in thesediagrams.

111. Interferometric Fan Traces

The interferometric fan trace plotter makes it pos-sible to measure the combined effect of aberrationsthat are present in the geometric optic fan ray tracesand the aberrations that are caused by manufactur-ing tolerances. According to Eq. (6), the setup inFig. 1 records the parametric family of functions

z = (f/a)[(n + 'A)X + AV-)=YJ]

n = 1, 2, .+3 ....

(13)

The fact that this plotter records a family of func-tions allows a convenient calibration of the system;according to Eq. (13) the distance z' between twoneighboring curves corresponds to a change of lX;accordingly, a change of 1X in AV is of the samemagnitude. With this calibration, the plotter rec-ords directly the difference of the wavefront at twotranslating sample points that are separated by thesampling distance a.

In the following interferometric fan traces of theFourier transform lens, the coordinate designationsare identical to those of Fig. 4, with the entrancepupil containing the pinhole aperture. The sam-pling distance a was chosen to be 10 mm, and thelight illuminating the pinhole aperture was from aHe-Ne laser.

The plots in Fig. 5 indicate that the wavefrontstays within the hX/4 limit in spite of the relativelylarge tolerances that were allowed for the manufac-turing process. The most striking feature of thesetests occurs in the plot YE = 0;zi = 0, where the fantraces show the largest deviation from theory. It islikely that a poor alignment of the two Fourier trans-

OBJECTAT -c

ENTRANCE IMAGEPUPIL PLANE

Fig. 4. Geometry for geometric-optical ray fans.


nnr(UJU

W1 IiU L Ir-

YE-0 21 o

-52mm ZE 52mm

YE .0 ZI 26 mm

-52mm ZE 52mm

YE *26mm Z1.O

-52mmZ 52mm

YE'26MM Z1.-26mmn

-52mnm -ZE 52mm

YEz52mm ZI VO

- -..~

I I I

- 2mm ZE 52mm

YE 2mm Z, '26mm

---

L- I

-52mm - ZE 52mm

YE *° Z, *52mm

I. I -52 mm - ZE 52mm

VE26mm Z1 .52mm

-..----.----

-52mel

- ZE 52mm

YE 52mm Zi 52mm

- -

2 I I-52mm -Z-w7E 52mm

Fig. 5. Interferometric fan ray traces through Fourier transform lens.

form lens triplets, which are in separate holders,onto a common axis accounts for this deviation.The discontinuous fine structure in these plots,which appears especially in the section YE = 52 mm;Z = 0, is partially caused by insufficient spatial fil-tering of the illuminating light. However, irregulari-ties in the refracting surfaces show a similar pattern,which must appear twice in each plot.

IV. Alignment Effects in the Fan TraceInterferometer

Owing to the principle of operation, a lateral opti-cal misalignment of the interferometer is not possi-ble; thus, the misalignments that must be consideredare longitudinal misalignments of the input and out-put film. This treatment then simultaneously pro-vides information about the effects that occur when

input film plane or output film plane are not perpen-dicular to the optical axis.

A longitudinal displacement P1 of the input signalf(z,y) from the front focal. plane results in a signal inthe front focal plane, which may be computed by

fq(z Y)= * exp( 2 Z2 +Y ), (14)

where * indicates convolution.Likewise, the output signal Fu(w,,wy) in the out-

put film plane which is displaced by 2 from theback focal plane is expressed by

Fa(w, y) = F(wz, wy)* P2 exp(-j 4f WP2 ) (15)

where F(wzwy) represents the light amplitude in the


. .I

back focal plane. Therefore, the effective inputfunction in the front focal plane is described by

f/(Z,fY) = 56(z - z- b - T)6(yb=O

-yi> exp[-j A V(zIy)]*p- exp( 2 A2 Y )

E pl exp[ iAV(z, + b + TY J

x exp[_j2(z - z- b -2 + (y -1 )] (16)

(b = O;a).

Lens L performs a two-dimensional Fourier trans-form of this function:

F(w2, w,)= jX exp[-j 4(wz2 + w2) exp{-i V(zb=O

+ b + T y) - [wz(z + b + T) + wyy]} (17)

b = O;a.

If, in addition, the final film is out of focus by adisplacement P2, the amplitude distribution on theemulsion becomes

that is well known to the lens designer, who utilizesthis skew property to select the focal point that com-pensates most effectively a given aberration configu-ration. This skew effect, therefore, makes the fantrace interferometer a convenient and extremely sen-sitive tool to select the best focal point for any par-ticular fan. The skew, which is caused by a dis-placement of the output plane, remains uneffectedby a simultaneous displacement of the input plane.However, if both planes are displaced, the completeset of functions is multiplied by a scaling factor [1 +P1P2/P2]. Since the aberration function AV(T), aswell as the term representing the spacing betweenadjacent curves, are multiplied by the same scalingfactor, the calibration property of the family of func-tions remains uneffected; therefore, the spacing be-tween adjacent curves represents a wavefront differ-ential of IX between the selected sampling points,separated by the distance a, and this property iscompletely independent of the location of input andoutput plane-as long as the Fresnel relations in'Eqs. (14) and (15) hold.

Equation (20) also indicates the temporal coher-ence requirements for the fan trace interferometer.Since adjacent curves are being used for calibrationpurposes, the location and definition of these curvesshould be unaffected by temporal coherence effects.

1j W 2 +W w 2 Xf2\F (w, Y) = F(W, Wy)* exp(_J P 4w ]

4wP + 2 exp[-~fp (WZ2 + W )2]piP2 + f 2 ex 1-4wp2

__ F2w A7r[z 1 + b + T - (2Xf2/4wP2 )Wzl2 + [Yi - (2Xf2/4wp2)wy1]2expLX(zl + b + ryl) - Pi + f2/p2

b = ;a,

and the recorded intensity pattern amounts to

= (PP + [2)2 {2 + 2 cos A AWT)Y=YI(AP2 + f2T2 I IT ~ 2fr p(2z, + a + 2T)- 2f2

+ A k a P1P2 + f2 ] (19)

If P2 = 0, the argument of the cos-function is identi-cal with that of Eq. (4). Thus, Eq. (19) reveals thefact that the fan trace interferometer is totally inde-pendent of alignment errors in the input plane if theoutput plane position is chosen correctly. Te argu-ment of the trigonometric expression in Eq. (19) de-fines the family of functions that results when onlythe minima of Eq. (19) are recorded.

= I + PP2)[V(T)- + )]

+ Z, + T + ) (20)

This equation indicates that an error in the outputplane location only (p = 0; P2 0) results in anidentical skew of each function in the family-a fact

For laser light, of course, the coherence propertiesof the light source far exceed the coherence require-ments of the fan trace interferometer, so that laserlight may be used without reservation in this case,where a Fourier transform lens is to be tested. How-ever, in testing a general imaging system, it is oftendesirable to use a source of partial temporal coher-ence for the purpose of ambiguity suppression.Since only one adjacent curve next to the zero ordercurve must be recorded, we may postulate that thiscurve-n = 1 in Eq. (20)-may be shifted by X/20 onaccount of the wavelength spread AX. This require-ment leads to

AX/X = (1/30)(a/f). (21)

Thus, the aperture ratio f/a of the fan trace inter-ferometer determines the tolerable wavelengthspread in the vicinity of the center wavelength X.

The spatial coherence requirements, on the otherhand, may be computed by virtue of the convolutioneffect that results in a diffraction pattern, when apoint light source is replaced by a light source of fi-nite extent. The net effect of this convolution pro-cess is a reduction in amplitude of the trigonometricfunction in Eq. (20). The value of the wavelength of


(18)

this function is X(f/a). This value also representsthe critical value for the projected dimension of thelight source (with projection center in the pinholeaperture plane), since it reduces the amplitude of thetrigonometric function to zero. A useful thumb rulefor the maximum dimension of the light source inthe direction of a line connecting the two pinholes,therefore, is (1/10)X(f/a). Since there is no func-tional limitation to the extent of the light source inthe perpendicular direction, a slit source may beused to illuminate the fan trace interferometer.

V. Fan Trace Interferometry in the GeneralImaging System

In the treatment in Secs. II, III, and IV, the Fouri-er relationship between corresponding planes provid-ed the key for the successful application of the fantrace interferometer. The general imaging system,e.g., microscope, photographic objective, or the tele-scope, usually does not provide an exact Fourier re-lationship between any pair of planes; however, thereexists an approximate Fourier relationship between anumber of planes, for instance, between the exitpupil of the general imaging system and its imageplane. In the following treatment the limits for theusefulness of the Fourier transform in the context offan trace interferometry will be computed. Withinthese limits the results from Secs. II, III, and IV maybe applied in toto.

If a spherical wave penetrates a diffracting apert-ure f(r) with vector coordinate r (Fig. 6), the lightamplitude u(a) in the focal plane (with vector coor-dinate ') is described with the help of the Huygen-Fresnel principle by

Ups)= f f(r) exp[-j(2-7r/X)(Il

+ r - 1 + r - a0)]idR.(22)

Since both planes containing the vectors r and oi areperpendicular to the axis, Eq. (22) takes the form

u(U) = ff(r)! expl-j(2r/X)(JlJ 2 + 1rJ2)1/2

- [1112 + Jr12 + 012 - 2(r ).]1/2ldR. (23)

A series development of the square roots up to thefourth power in ai and r leads to

u(cr) = f f(r) expt-j(27r/X)[(rY)/'1I - (1cT12 /2111)11.

exp - A

11 lull+ 4(ru)2 + 2JrJ21- 4ra[Irl + U1I2]})dR.(24)

The first term of the exponential expression repre-sents the term indicating the Fourier transformproperty between the two planes. The second expo-nential term, which does not affect the value of theintegral, represents a phase pattern indicative offield curvature. The third term may be looked uponas the error term that defines the limits of the Fouri-

er transform property in this setup. A useful ap-proximation for the largest phase error 4'max, whichoccurs when o. and r are 180° out of phase, is givenby the following expression:

(25)

which indicates the phase error in terms of the semi-field angle and the semi-aperture angle, up to whichthe fan trace interferometer can be used in thissetup. For instance, if the maximum allowable error4)max is X/20 and the aberration function family ofthe fan trace interferometer is recorded along an in-terval of I = 20X, the semiaperture value computesto 0.108.

In order to avoid any ambiguity in the fan tracerecord, the inferferometer should be designed suchthat either the useful range (I o = 20X) of the fantraces is actually recorded or that this range is prop-erly outlined so that it may be identified. This re-striction may be accomplished either by a properchoice of the pinhole diameter or by a proper choiceof the illuminating light frequency bandwidth.

The limitations of the Fourier transform geometryin Fig. 6 which are expressed by Eq. (24) may alsobe determined by pure geometrical considerations.

If a diffracting aperture A - B (Fig. 7) is illumi-nated by a spherical wave converging toward C, aparticular spatial frequency induces the same dif-fraction angle a to all rays. Therefore, the points A,B, C, and D must be on a common circle. This ex-plains the field curvature for this Fourier transformsetup. However, the corresponding points of differ-ent rays, e.g., the rays EC and FC and their by adiffracted rays, lie on a different circle. Therefore,there is also an aperture limitation, which corre-sponds to the limit indicated by Eq. (25).

Fortunately, the general imaging system containsseveral planes that may be used to perform Fouriertransforms. In Ref. 5, it was pointed out that theideal Fourier transform lens is situated in a telecen-tric ray system, where the ray convergence in theback focal plane is aplanatic as well as anastigmaticand where the chief rays satisfy the sine condition.A photographic objective ideally satisfies only therequirements of aplanasy and anastigmatism in theback focal plane, a fact that restricts this lens as aFourier transform system to field angles within thelimits of Gaussian optics. However, unlike the con-verging light Fourier transform setup, there is no re-striction on the aperture angle in this case; therefore,

i I a

Fig. 6. Fourier transform geometry for a spherical wave.


7rIrI4)max = 8 - I I�1'1 +

xll� III 'Jily

A 0

J3

Fig. 7. Geometry of a diffracting aperture.

for the purpose of fan trace interferometry, the Fou-rier transform is best performed if the input functionis located in plane parallel light. This test scheme isalso desirable from a practical point of view, since itis relatively simple to illuminate, for instance, acamera objective with two partially coherent lightemitting pinholes (Fig. 8) and to record the slit imagevia a stationary relay microscope onto the outputfilm. The alignment requirements are identical tothe results in Sec. II; therefore, if the output slit, ora virtual image of it, is in the back focal plane, theaxial position of the translating pinholes may bechosen at any convenient location. And, within thevalidity of the Fourier transform, all results fromSecs. II, III, and IV apply.

The limits in this case are caused by the fact thatthe photographic objective is free from distortion.Therefore, these limits may be computed by amethod identical to the one that leads to Eq. (12).For instance, for a photographic objective of 50-mmfocal length, the useful recording range I I for fantrace interferometry is 2000X in the vicinity of theaxial point in the output plane. Furthermore, sincethe ideal photographic objective provides a flat fieldin the output plane and since the cosine law6 postu-lates the fulfillment of the sine condition for off-axispoints as well, this identical large range of 2000Xapplies for all points in the output plane. To makeit valid, it is necessary to incline the pinhole illumi-nating optics, so that the off-axis point in questionreceives zero-order diffracted light.

The limitations mentioned so far, namely the ap-erture limitation in the converging light setup andthe field angle limitation in the parallel-light setupof a distortion-less lens, are not limitations in thesense that they form an absolute limit for the useful-ness of the fan trace interferometer. For instance,Eq. (23) may be computed exactly for any aperturewith f(r) substituted by the mathematical expres-sion of the pinholes, in spite of the fact that the Fou-rier transform does not hold for large apertures. Theresult will be that the calibration lines of the fantrace interferometer are curves rather than straightlines. Likewise, the diffraction pattern of two pin-holes may be computed after being transformed by aphotographic objective without distortion. Here, theresult will be that the hitherto constant spacing ofthe calibration lines becomes nonuniform. In eithercase, however, it is possible to compute the expres-sion that results when the wavefronts are ideal, anda deviation from that pattern due to aberrations canbe analyzed qualitatively as well as quantitatively.

Here, as in the case of conventional interferomet-

ers, it is necessary to first compute the pattern of anaberration-free wavefront and second to measure thedeviation from that pattern. However, as long asthe fan trace interferometer stays within the limits ofthe Fourier transform validity, the interpretation ofthe resulting pattern is extremely simple; in thiscase, the ideal pattern is represented by straightlines of equal spacing, where the calibration is suchthat the distance between adjacent lines represents awavelength differential of iX between two samplepoints of the wavefront. Therefore, it is possible toread off the wavefront aberrations without computa-tion and without a calibration procedure. This isalso true in the case of the photographic objective ifit is to be tested for finite object distances. In thiscase the values p and P2 in Eq. (20) represent con-jugate distances, which may be expressed with thehelp of Newton's formula. The recorded pattern inthe output plane becomesy = (2f/la)[AG-r)y=,, - (n + 2 )X]

+ (f/P)i Z, + T + (a/2)]. (26)According to this equation, the necessary displace-

ment 1 will merely cause a tilt of the completefunction set. However, the lines remain straight ifthe lens is aberration free for the chosen object planelocation P1. It is not necessary to measure the dis-placement P2 since it may easily be computed fromthe tilt of the record and from the last term of Eq.(26).

Analog considerations apply for the fan traceinterferometric testing of a microscope. Since thelight waves emerging frorm the eye piece under nor-mal operating conditions are plane parallel, or nearlyso, the microscope is best tested in reverse, with thedouble pinhole aperture illuminating the eye piece(Fig. 9).

If the equivalent focal length of the microscopeunder test is very short, the fan selection by the hori-zontal slit is best performed in a location other thanthe focal plane, either inside the relay microscope orbehind it. The quality of the relay microscope doesnot necessarily have to be better than that of the testmicroscope; for instance, a certain amount of distor-tion in the relay microscope is not objectionable,since the spacing between adjacent curves on therecord provides a continuous calibration along theordinate in the output plane. The pinhole spacing a

LIGHT PINHOLE SLIT OUTPUTSOLRCE COLLIMATOR APERTURE APERTURE FILM

Fig. 8. Test setup for camera objective.

July 197.3 / Vol. 12, No. 7 / APPLIED OPTICS 1649

COLLIMATOR MICROSCOPE OUTPUTTO BE TESTED FILM

LIGHT PINHOLE SLITSOURCE APERTURE APERTURE

Fig. 9. Setup for interferometric microscope test.

must be chosen small enough so that the test micro-scope as well as the relay microscope can resolve thefringe interval (f/a)X. The additional resolutiongain, which results on account of overexposure andoverdevelopment of the output film, is independentof the microscopes involved, provided that they re-solve the fringe interval. This gain in resolution maybe in the order of (1/50)(f/a)X for the methods men-tioned, but, it may be pushed to a value of (1/200 f/a)X by a photographic process, which recordsonly the lines of equal density.7

The chromatic behavior of the fan trace interfero-meter may also be analyzed with the help of Eq.(20). Since the last term does not contain the pa-rameter X, the tilt of the function set, which iscaused by a finite value of p2, is not affected by lightfrequency changes. However, the interval betweenadjacent fringes changes with X, so that the calibra-tion of the interferometer is always in units of onelight wavelength at the selected light frequency.

VI. Conclusions

The fan trace interferometer has been found to bean effective and accurate tool in the practical evalu-ation of the performance of a Fourier transform lensand in the measurement of its aberrations. Thislens type generally exhibits a short back focal dis-tance, which makes it difficult to apply the shearinterferometer for the purpose of wavefront analysis.The fan trace interferometer requires only the inser-tion of a double-pinhole aperture into the ray path

between the light source and the Fourier transformplane. The axial location of this laterally translat-ing aperture may be chosen within a wide rangewithout altering the fan trace record. This recordcontains a set of identical functions with a well-de-fined spacing between adjacent curves, which pro-vides an automatic and accurate calibration of thefan trace interferometer. Furthermore, each func-tion of the set exhibits the familiar fan ray tracetrack (with interferometric resolution). Therefore,the interpretation of the record and the identifica-tion of individual aberrations as well as their mea-surements are greatly simplified.

Due to the wide freedom from alignment require-ments, this interferometer has also served an ex-tremely useful purpose in the investigation of com-plete coherent optical systems, inclusive collimator,mirrors, windows, etc.

Since all optical imaging systems, coherent as wellas incoherent, contain planes, where a Fourier rela-tion holds within certain limits, the fan trace inter-ferometer may be applied for the purpose of systemevaluation. Within those limits, all results pertain-ing to calibration, interpretation, accuracy, andalignment requirements apply. However, the inter-ferometer may also be used in situations where thelimits of Fourier transform validity are too small tobe applicable. Here, the aberration free patternmust be computed and the comparison of the fantrace record with this ideal pattern provides the nec-essary aberrational information.

This paper was presented at the Annual Meetingof the Optical Society of America, San Francisco,California, October 1972.

References1. A. A. Michelson, Philos. Mag. 30, (5), 1 (1890).2. Lord Rayleigh, Proc. Roy. Soc. (London) 59, 198 (1896).3. F. Twyman and A. Green, British Patent 103,832 (1916).4. W. J. Bates, Proc. Phys. Soc. (London) 59, 940 (1947).5. K. von Bieren, Appl. Opt. 10, 2793 (1970).6. M. Herzberger, Modern Geometrical Optics (Interscience, New

York, 1958), Vol. 8, p. 160.7. H. Schmidt, Erfahrungen mit dem Schwarzungsplastikverfah-

ren, Optics, 16, 538 (1959).

Donald L. WebsterBausch & Lomb


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wavefront investigation of a fourier transform lens with the fan trace interferometer

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