Diffraction Effects of a Focal-Plane Slit Shutter and ItsInteraction with Uniform Linear Image-Motion

Lucian Montagnino

A general expression for the exposure transfer function of a photographic system employing a focal-plane

shutter is derived which includes the effects of uniform, linear image-motion. The results show that the

modulation transfer function (MTF) of such a system is upper bounded by the product of the diffraction-

limited optical MTF and a term strongly coupling the shutter and motion effects. The coupling of these

effects results in a general solution dependent upon the size and shape of the pupil and the aberrations

of the optical system.

Introduction

The objective of this analysis is to formulate a mathe-matical model that predicts the performance degrada-tion of a photographic system due to the diffractioneffects of a focal-plane shutter and its interaction withimage motion. Since image motion is a primary sourceof image degradation in photographic systems, the factthat it is coupled with the shutter effects adds signifi-cance to shutter performance considerations.

Specifically, a general solution for the exposure trans-fer function of a system employing a focal-plane slitshutter is derived that includes the effects of uniform,linear image-motion.

It should be noted that this is the most common typeof focal plane shutter, and it is generally considered tobe the fastest type of mechanical shutter. Hence, it isexpected that such a shutter minimizes image motioneffects. However, shutter diffraction effects can negateimage motion gains if they are not given careful con-sideration.

AnalysisSince photographic film responds to energy, the

quantity of interest is the time-integrated optical imagewhich is referred to as the exposure image.' Assumingno reciprocity failure in the photographic material, theexposure image is an intermediate stage between theoptical image and the statistical latent image. Thestatistical latent image is obtained from the exposureimage by a nonlinear transfer. Since the exposure im-age is obtained from the optical image by a linear pro-cess, it is the logical termination point for an analysis in-volving time dependent diffraction effects.

The author is with Perkin-Elmer Corporation, Optical Tech-nology Division, Danbury, Connecticut 06810.

Received 26 October 1971.

Derivation of Exposure Transfer Function

Mathematically, the exposure image is given by

=tE,)= J ~~)(t;~t (1)

where

I[t(t),7(t);t] =optical image, i.e., intensity (power)distribution;

E~t~x7)= exposure image, i.e., energy distribu-tion;

(t) = instantaneous coordinates of an arbi-trary image point;

= coordinates of an arbitrary point inimage plane.

Since the operations considered are linear, the processesleading to the exposure image can be characterized by atransfer function. Physically, however, the impulse re-sponse is the observable quantity of interest. LettingI[t(t),7(t) ;t] denote the intensity impulse response andE(t,-q) denote the energy impulse response, one can pro-ceed to formulate an expression for the exposure imagetransfer function.

First, consider the effects of the shutter. Physicallythe slit determines the exposure time for each pointimaged. For a given point one can readily see that thefocal cone is vignetted by the slit in the exposure process(see Fig. 1).

Since mechanical considerations necessitate that theslit lie a considerable number of wavelengths forward ofthe focal plane in general, vignetting of the cone resultsin a far field diffraction effect. Hence, the slit can beprojected back to the pupil and thus can be thought ofas a pupil shutter.

Mathematically, the pupil function is modified by awindow function which is a projection of the focal-planeslit into the pupil plane. Noting that the slit is pro-

926 APPLIED OPTICS / Vol. 11, No. 4 / April 1972

right. The exposure image is arrived at by substitutinginto Eq. (1) which after rearranging terms becomes

dxdy JJ dx'dy' f(Y)f*(xtY')

Fig. 1. Schematic of slit shutter geometry and operation. W =slit width; h = separation distance between slit and focal plane.

jected back from the focal point, the expression for theintensity is given by

I[4(t),t) ;t] = f+o.

X exp{i(27r/Xf)[(x' - x) + i(y' - y)]}

x f dtp(x + vt)p(x' + vt) X exp{ -i(27r/Xf)

X [t(x - x') + vyt(y - )]

Note that in general the integrands considered arebounded functions; hence, changing the order of inte-gration is permissible. Fourier transforming the expres-sion for the impulse response yields the unnormalized,exposure-image transfer function,

D(u,v) = ff dtd,1 exp[i2r(u + vr7)]E(%,,). (3)

f(x,y)p(x + Vt)

X exp{-i(27r/Xf)[x4(t) + y(t)]}dxdy| (2)

where

t = time coordinate;x,y = pupil plane coordinates;

X = wavelength;f = focal length;

f(x,y) = pupil function;p(x + vt) = window function for a slit moving in

the negative x direction{p(z) = 1 IzI < d/2I = 0 otherwise;

vs= pupil shutter velocity;d = projected slit width

= (f/h)w= Vste;

w = slit width;h = separation distance between focal plane

and limiting edge of slit (see Fig. 1);te = effective exposure time.

For uniform, linear image-motion, let

(t) + vt,7(t) = 1 + vAt,

where v and v are the image velocities in the x and directions, respectively. Substituting into the expres-sion for the intensity and expanding yields

I11W),r7(*)t] =fJ dxdy ffJ dx'dy'(f (,y)f *(x',y') /

X p(x + vt)p(x' + vt) exp{i(27r/Xf)[4(x' - x) + (y' - y)]}X exp{-i(27r/Xf)[vxt(x - x') + vt(y -y')]I),

where * denotes the complex conjugate, and the inte-grals are written in operator form, i.e., operating to

Using the theory of generalized functions, the transformintegration yields a two-dimensional Dirac delta func-tion, i.e.,

ff dtd,7 expti2i{4(u + X7 ) + { + Xf= (Xf)'(x' - + Xfu)5(y' - + Xfv)

where 8( ) denotes the delta function. Performing theintegration on the x',y' and time variables, respectively,yields the expression for the unnormalized exposuretransfer function

D(uv) = t)(Xf)2 exp[iilI. UQf~u] in[7rl . u(u)]

X f ' dxdyf(x,y)f*(x - fu,y - fv)exp[+i2ir(- )x],

where

~(u) 1 - (Xf/d)Iul for ul < (d/Xf)= 0 otherwise,

and

I u = Iu + lv,

with 1,1l denoting the image displacements in the x andy directions, respectively.

Assuming a circular pupil boundary, let us nownormalize the pupil coordinates as follows. Let x =(2a) a and y = (2a)#3, where a = radius of pupil. Also,let us normalize the spatial frequency to the optical cut-off by the substitution u = (1/XF)v, where X = wave-length and F = f number.

Using the same function symbols, substitution yieldsthe normalized exposure transfer function

April 1972 / Vol. 11, No. 4 / APPLIED OPTICS 927

En =Y.

Explicitly, the general expression for the exposure trans-fer function is

,r~v.~,vy ex 71 v\ P.,] sin[7r(l - v/XF),D(v.,)]

vz = expL ~r(!if) n 7r(1 v/XF)

CJ dadflf(ce,)f*(a - v.,3 - v) exp[+i2ir(I v/xF)a/vo]

X ~~r+¢J J-c dadfl |f(,3)| 2

(4)

where

vx,yv = normalized x,y-direction spatial frequencies,respectively;

-1 < v 1-1 V v, • 1;

V < a < ;- I-2< < ;

1 = vector displacement of image;vo= (w/h)F;F = f number;w = slit width;h = separation distance between focal plane and

limiting edge of slit;

and

1 - (I |I/vo) for I v. I < v,

= 0 otherwise.

Limiting Cases

Consider the following two special cases.

Case 1

Shutter effect with no image motion. Mathemati-cally, this means III = 0, hence, the transfer functiontakes the form

Irr:

Case 2

Image motion with no shutter effect. Mathemati-cally, this means vO -- , or in terms of physical vari-ables, the slit width-to-separation distance ratio ap-proaches the limit (w/h) - -. Performing the limitprocess, the expression for the transfer function becomes

sin[7r(l v/XF)]M(V2Vy) = (i/X) 0(v-,P)

where ro(vf,vy) is the optical transfer function as definedin Case 1. Note that the image motion effect isseparable and is the standard result found in the litera-ture. Examination of the motion term shows that ithas no modulation effect in the direction orthogonal tothe image displacement.

General Performance Limitations

Examination of the general solution shows that imagemotion and the shutter effect are coupled to each otherand thus to the pupil function and as a consequencecannot be represented by the product of independenttransfer functions. Specifically, the presence of a phaseterm inside the pupil integral causes their nonseparabilityfrom the optical characteristics. The fact that thiscoupling results from a phase term, however, enables usto upper bound performance. Expressing the pupilfunction in terms of its amplitude and phase com-ponents,

f(ceg) = a(a,o) exp[ik(a,3)],

Schwarz's inequality 2 can be applied to the general ex-pression for the exposure transfer function to yield

(6)| Sin[7r(1 /XF)4,(v.)]I T(V2,"V)I < I i (I v/XF) roo(vx,"y),

where ro0(P:,vy) and ](v) denote the diffraction-limitedoptical and shutter transfer functions, respectively.The inequality provides an upper bound on performancein terms of shutter parameters and image displacement.

dadftf(cef3)f*(a - .,f - O

ff2: dadgljf(,3)I 21.0

Note that the shutter effect is separable and that theterm

ff2 dad3f(e,I3)f*(a - v,3 - v)

fT ., dadfIf(aS3)I 2

is the optical transfer function. Thus, one can define ashutter transfer function'

b(v.) = 1 - (I P.I /"o) for vI < Po

= 0 otherwise,

0.8

Z

z 0.6

0.4

0.2

(5)

which it should be noted does not affect the y direction.The result for no image motion can be rewritten as

rs(z'Vx,2 ) = -XP-)To(P-,Pv).

0.0

x= 3.0

= 3.0

Vx0.0 0.2

NORMALIZED SPATIAL FREQUENCY

Fig. 2. Rlativo performance limitations due to shutter/imagemotion interactions.

928 APPLIED OPTICS / Vol. 11, No. 4 / April 1972

.

Ts.(V.,V8) CVas

Tro(V.,vV)=

To illustrate the significance of this upper bound onperformance consider

I (V2,0) < sin[7r(l2/XF)v-4(v 0 )]roo(vxO) 7r(I.1XF)v.

with image motion in the amount of lz/XF = 3.0 for theconditions = 3.0 and vo -- o (ideal shutter). As canbe seen in Fig. 2, the results for the two conditions arequite different.

Directional ConsiderationsReturning to the general expression for the transfer

function, let us now consider directional effects. Theexpressions for the transfer functions in the x and ydirections are

T = r( l vz21 sin[ir(,/XF)v-1(v-)],r(V2,+) = expr (l2\ "~ 1JJ dad3f(a,g)f*( - v) exp[+i2r -V- a]

XfT .' I - ~~~'XE )

dad3 f(,,3) 2

sin [7r(ly/XF)vzy]r(O v ) = 7r(1ydaFfay

JJ . dcdf(.,X -X

()f*(a,3 - Py) expF+i2 ( -I) -]L \XE POj

dad3jf(.,13) 2

and

ii't

T(x,O) for Diffraction-Limited Circular PupilSince modulation effects in the direction orthogonal

to slit edges are a primary consideration both opticallyand mechanically, let us calculate (vO) for a diffrac-tion-limited optical system with a circular pupil forillustrative purposes.

Letting 1(Qx) denote the shutter transfer function asdefined previously, and restricting the mathematicalexpressions to positive spatial frequencies, i.e., v > °,for mathematical expediency, one can derive a seriessolution. The result for the first few significant terms is

= sin[ir(Q./xF)v2(v.)J ( (, ,)

(r(.,0 7rAF) ) (0I 0A)))(J [1 - I(Vz)]} os XF-L)vzl1 (xI

+ 8 [1-X)(v)] sin(7r(k)vz[l -

X (1 -V2) - [1-(V)]

X cos Ir(-F)vxii - 4(Yx)]J [v(1 -

- COS '(Vz) - 1pV2(1 - V2)] + ) (7)

where rOO(vZO) is the diffraction-limited optical transferfunction in the x direction and JI }, J{ }, . . . etc.,are Bessel functions of the first kind and of order 0, 1,

., etc. For small degradations the terms of the seriescan be expanded to produce

respectively. Note that the shutter and image motioneffects are not separable in either direction.

Consider the following special cases.

Case A

Image motion in the x direction alone yields

T(I'xO) = r(v,O) (general expression)

and

r(OFP) = TO(0,2V1)-10=0

As can be seen, the y direction is unaffected/

Case B /

Image motion in the y direction alone yields

T(vP,0) = (iz)ro(vx0)

and

r(Op2/) =r(O,vy) (general expression).

10 0

In this case, the shutter affects both directions. Theimage motion does not affect the x direction, and as aconsequence the shutter effect is separable in that direc-tion.

sin[ 7r(1.1AF)vx-T(v.)]T(P.,0) = r(lx/XE)vx

X (roo(v,,0){1 - r2(l/XF)2[1 - p(vz)]2!(1 + Vp2)}+ (4/37r){7r2(lx/XF)2[1 - .(V.)]2}v_(1 -

- (1/2r){ 7r2

(1/XF)2

[1 - ,(Vx)]2} [v(l - V2)l - § COS-(v)

.1V2(1 - V2

)] + *-

where terms of the order 7r3(lx/XF)I[1 - 4(j') ]3 and

higher have been neglected. Note that the result isconsistent since for no image motion it reduces to

r(v.,O) | = (Px)roo(v,,0),

and for no significant shutter effect, i.e., 4.(vx) = 1.00it reduces to

sin[7r(1.,/AF)v.,]T(~0) 0 l/E~

r (IZ/AF)vx "x

as expected.

Solution for Diffraction-Limited Square PupilThe solution for a diffraction-limited square pupil is

quite illustrative since it exists in closed form. The

April 1972 / Vol. 11, No. 4 / APPLIED OPTICS 929

, 1 I _

jj--t'°

solution for the exposure transfer function is given bythe expression

Tr(vxPv) sin[7r(I v/Xf)b(v)]r~rzJ~v) =7r(l * v/xF)(8)sin[7r(I * v/X)(l/vo)roo(v)o

7r(1 * v/XF)(1/vo) ov)

where the diffraction-limited optical transfer functionis defined by

7-oo(V,Vy) = 7oo(p.),roo(v)

(1 -v.)(1 -| I)for "vI and Iv/I 1

0 otherwise,

and the other terms are as defined previously. Ex-amination of the result shows that the shape of thepupil is a significant parameter and that the solutionreduces to the appropriate limiting forms for the condi-tions of no image motion and no shutter effect as itshould. Also note that the shutter/motion effects areseparable in the y direction in contradistinction to thegeneral form.

ConclusionsA general solution for the diffraction effects of a focal

plane shutter and its interaction with uniform, linearimage-motion has been derived [see Eq. (4) ]. Examina-tion of the results indicates that the shutter/motioninteraction is dependent upon the size and shape of thepupil and the aberrations of the optical system, whereasseparately the shutter and motion effects are indepen-dent of these parameters. Further analysis, however,shows that one can upper bound performance in lieu ofan explicit solution:

sin[7r(l v/xF)b(v.)] (VXV2),

| r(V,^v~ < |7r(I v/XF) 110

where rOo(v,vy) and b(v) denote the diffraction-limitedoptical and shutter transfer functions, respectively.

References1. R. V. Shack, Appl. Opt. 3, 1171 (1964).2. A. Papoulis, Systems and Transforms

Optics (McGraw-Hill, New York, 1968).with Applications in

COVER

The cover shows fish-eye photographs of a laser beam reflected many times in-side a gold-coated spherical cavity. These cavities have been developed as longpath, far infrared, nonresonant absorption cells. The laser beams illustratesome of the optical properties of the spherical cavity as shown in Fig. 2 on page725 of this April issue. A wide angle camera made the photographs-taken byFriedrich Hoelzl at the National Bureau of Standards-Boulder-through a largeneck in the top of the cavity. The photograph in the bottom right-hand cornerwas Grand Prize Winner in the 1971 Institute for Basic Standards PhotographyContest at NBS. The color cover was provided through the courtesy of the

National Bureau of Standards.

930 APPLIED OPTICS / Vol. 11, No. 4 / April 1972