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Optica Acta: International Journal of Optics

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The Solution of Some Inverse Diffraction Problemsin Terms of the Diffraction Coefficients

M. Novotný

To cite this article: M. Novotný (1977) The Solution of Some Inverse Diffraction Problems inTerms of the Diffraction Coefficients, Optica Acta: International Journal of Optics, 24:5, 577-590,DOI: 10.1080/713819596

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OPTICA ACTA, 1977, VOL . 24, NO . 5, 577-590

The solution of some inverse diffraction problems in termsof the diffraction coefficients

M. NOVOTNIInstitute of Instrument Technology, Czechoslovak Academy of Sciences,Kralovopolska 147, 612 64 Brno 12, Czechoslovakia

(Received 19 March 1976)

Abstract. The form of the recently developed series representation of theaxially symmetrical diffraction fields enables us to establish the pupil functionfrom the series coefficients yk(u) . A new interpretation of the Strehl intensityratio is given. It represents the ratio of the energy coming to the image focusand that coming to the other foci of the filter . Further, by the use of the three-term approximation, the transmissivity of the filter giving the desired two-pointresolution and attaining the maximum value of the Strehl criterion is found andcompared with the exact solution proposed by Luneberg. Finally, it is shownthat the Luneberg filter transmissivity represents a common approximative solu-tion of the apodization problems that utilize the integral criteria of apodization .

1 . IntroductionSince the diffraction field G(u, v)-cf. [3, 7],

1G(u, v) = f g(t) exp (iut/2) J0(vt1 /2 ) dt,

0

represents an approximative solution of the Helmholtz equation for certainboundary problems [1, 2], one can expect that there exists an unambiguousreciprocal transformation of the diffraction pattern G(u, v), u = const for gettingthe filter transmissivityg(t) . The integral representation (1) generates the integ-ral form of the inverse transformation-the well-known Hankel transform ofthe zero order (cf., e .g., [15], p. 97) . In the series representation of G(u, v)derived in [3]

00G(u, v) = I (-1)kyk(u)/k(v),

(2)k=0

the transmissivity g(t) is involved in the diffraction coefficients yk(u), which havethe form ([3], equation (29))

1yk(u)= f g(t) exp (iut/2)Qk(t) dt.

0

(1)

(3)

We may then suppose that the pupil function g(t) can be expressed from the setof the diffraction coefficients {yk (u)}, k = 0, 1, . . . . The inverse transformationof the diffraction coefficients yk(u) to the original diffractorg(t) will be derived in§ 2 by replacing the functions Qk(t), which represent the renormed Bernoullipolynomials ([3], equation (24)), with the orthogonal Legendre polynomials .

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Whenever we are looking for the filter transmissivity whose diffractionpattern is required to have some prescribed property, we proceed to solve acertain inverse problem . The pupil function expansion in terms of diffractioncoefficients may be employed in a number of such inverse diffraction problems .These problems usually admit of no exact solution . Therefore, variousnumerical methods have been elaborated [13, 14] . The present approach isespecially efficient if the requirements are imposed on the diffraction field not farfrom the axis v = 0 . Then, a rapid convergence of the series (2), enables us torestrict ourselves to a few first coefficients yk(u) .

In its nature, the present method is similar to the previous procedures developedfor solving the apodization problems [13] . The apodizing transmissivity hasbeen usually found in the form of a linear combination of certain functions whoseHankel transformation is known . Thus, the coefficients of the combination aresearched . The efficiency of the methods depends first of all on the properties ofthe functions used . Unlike these linear combinations we expand the pupilfunction in the orthogonal set of the Legendre polynomials . Thus, the conver-gence of the pupil function expansion in the sense of the least squares fitting isguaranteed .

In the present approach we express the diffraction field in terms of the specialfunctions /k(v)-cf. (2) . They have a number of useful properties-cf . § 5 in[3], in particular, they satisfy (k = 0, 1, 2, . . . )

dfdv(v) = 2d k(v)- (4)

From [3], equation (17), it follows that their asymptotic behaviour is characterizedby the expression (k = 1, 2, . . .)

~,

1

v 2k-2d k(V) 2t (k- 1)! (/2)

for v-_> oo .

Thus, the/k(v) are unlimited and cannot then have any Hankel transforms whichare suitable when calculating integral intensity characteristics-cf. [13], p . 65 .This disadvantage can be overcome by using the Parseval theorem-cf . §§ 4and 5 .

We illustrate the present approach of solving inverse problems in terms ofdiffraction coefficients yk (u) by using as an example one of the Luneberg problems .In § 3 the formulation of this problem of finding the filter transmissivity with thedesired value of two-point resolution and the maximum Strehl criterion is dis-cussed and modified. The exact solution 1(t) is found to be a common approxi-mative solution of three apodization problems (§ 4). Section 5 presents thesolution of the problem discussed in § 3 in terms of the diffraction coefficients .

2. The pupil function expansion in terms of the diffraction coefficientsFor simplicity let us set at first u =0 and write briefly yk(0)- yk . In agree-

ment with (3) the coefficients represent certain scalar products of the pupilfunction g(t) and the Bernoulli polynomials Qk(t), i .e . yk = (g, Qk) . If the setof polynomials {Qk(t)} formed an orthogonal basis, the pupil function g(t) couldbe simply expressed by means of yk . Unfortunately, the Bernoulli polynomials

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are not orthogonal and we must then use another polynomial basis which isorthogonal with respect to the considered scalar product .

The orthogonal basis of the space of the functions, which are defined in theinterval <0, 1> and satisfy Dirichlet's conditions, is formed by the set of theLegendre polynomials p8(x) transformed and renormed in an appropriate way

The Legendre polynomials can be generated using Rodrigue's formula ([4],p . 267)

1 d8(x2 -1)sp8(x)= 23s! ' dx8

or by the use of the recurrent relation (s +1 )p 8+1 (x)+sp s_ 1(x)=(2s+1)p 8(x)with p o(x)=1, pl(x)=x . By means of the transformation x = 2t - 1 we attainthe orthogonality of the functions (5) in the interval (0, 1>

Inverse diffraction problems

579

s!P8(t) _ (2s)i p8(2t -1) ; s = 0, 1, . . . .

1(Pr, Ps) = f Pr(t)Ps(t)dt=wa bra,

(6)0

s!

2

1WS=

[(2s)!] 2s+1'

The first polynomials P,(t) are of the form

P0(t)=1, P1(t)=t-1/2,

P2(t)=t2/2-t/2+ 1/12,

P3(t) = t3/6 - t2/4 + t/ 10 - 1/120 .

Comparing with the polynomials Q8 (t) ([3], equations (25)) we see that the firstthree polynomials P8(t) and Q 8 (t) coincide .

Let us study the transformation matrices {(X,r} and {p s r} relating the sets{P8(t)} and {Q,.(t)} :

~r

a

Qr(t) = L. am'•Pm(X), P8(t) _

NnsQn(t) .

(9)M=0

n=0

They evidently fulfilrL, 0mrNnm= srn) r > n,

(10)m=n

8Q samn=3Nn

sm'n=m

(5)

(7)

(8)

s>,m .

(11)

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Thus, knowing, e.g., matrix {a,,, r } the relations (10) and (11) enable us tocalculate successively the inverse matrix elements 3 fly Nnn+l, . . ., /3 r or re-spectively, fs8,

8 . .

8S

Any element a,,,r may be evaluated by means of the relation

amr=Wm-1(Qr, PM)

following from (9) and (6) . The normalization of the polynomials (5) leads tothe unit values of all diagonal elements of both the matrices, i .e .

arr =1, Prr=1 ; r=0, 1, 2, . . . .

(13)

Further, because of the zero mean value of the polynomials Qr(t), i .e .

1

f Q r(t)dt=0, r=1, 2, . . .,

(14)0

following from their recurrence property ([3], (24)), the relation (12) gives form=0

a0r=0 r=1,2, . . . .

(15)

Thanks to certain symmetry properties common to the polynomials Qr (t) andPr(t) of the same degree r, further elements amr must be zero . From the recur-rence relation ([3], (24)) and Rodrigue's formula it follows that both the poly-nomials Q r(t) and Pr (t) are odd (even) with regard to the axis t= z of the absicissa<0, 1) if the degree r is odd (even) . Then, because of (14), the scalar products(12) are zero whenever Qr (t) and Pm(t) show different symmetry . Therefore,

amr=0 if r+m=1, 3, . . . .

(16)

With regard to (9) it holds also that

Nmr=0 if r+m=1, 3, . . . .

(17)

This property facilitates the calculation of the inverse matrix {N8r} from (10) or(11). Let us write out the elements o4 Br and /3,r of the lowest indexes

a0 0

al1 a0 1

a22 a1 2 a O 2

a33 a23 0613ao 3

1

1 0

1 0 0

1 0 6O 0

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Similarly,

O.A.

{Rsr} =

Inverse diffraction problems

581

1 0 so 0

Knowing the inverse matrix {p8r} it is easy to derive the unknown functiong(t) from the diffraction coefficients y o, yl , . . . . Let us expand the function g(t)in the orthogonal basis (5)

0

Inserting the expression (9) for the polynomial P8(t) into (21) and using (3) withu = 0, we get P 8 in terms of the diffraction coefficients

80

8

C8 Q

re=8-1 g, I NnsQn - Ws-1 G Nns yn-n=0

n=0(22)

Hence, the expansion (20) together with (22) enables us to evaluate the pupilfunction g(t) from its diffraction coefficients yn .

By the relation (22) the set of the first N diffraction coefficients yn determinesthe first N coefficients 17 0 , F1 , . . . , P N _ 1 of the expansion (20) . Therefore, restrict-ing ourselves to solving a problem to the first N diffraction coefficients {yk },k = 0, 1, . . . , N - 1, the corresponding N-terms pupil function expansion in the setof the Legendre polynomials (20) represents the best least square approximation .

If we want to establish the pupil function g(t) from its diffraction coefficientsyn(u) associated with any Fresnel diffraction pattern (u # 0), we can use the samerelations (20), (21) and (22), replacing however the polynomials P,(t) and Qn(t)by the functions Pe(t) exp (- iut/2) and Qn(t) exp (- iut/2) . The invariant formof (21) and (22) is a consequence of the fact that the new basis {P8(t) exp (- iut/2)},s = 0, 1, . . . satisfies the former orthogonality relation (6) and that the trans-formation matrices in (9) remain unchanged .

3 . The superresolving filter 1(t) attaining the maximum Strehl criterionThere is an infinite number of the filter transmissivities giving the desired

performance property . Every improvement of one property goes however atthe expense of another complementary performance ability of the filter . Thus,the reducing of the width of the focal intensity maximum implies an increaseof the brightness of the fringe system . The two-point resolution is then improvedat the expense of inferior resolution of the high contrast details . We are interestedin such filters that display the desired performance ability and moreover are thebest in regard to other important performance properties . Such a formulation

2Q

g(t)=

r8P8 (t), (20)8=0

1r8=w,i f g(t)P,,(t)dt=w,-1(g, P 8 ) . (21)

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leads to variation problems . We shall discuss in this section such a problem offinding the super-resolving filter attaining the minimum energy in the fringesystem .

A useful relative criterion of controlling the total energy in the fringe systemrepresents the quantity (cf . [5])

str [g(t)] = I f g(t) dtI 2/ .j jg(t)j 2 dt

(23)

introduced originally by Strehl [6] . The definition of the Strehl intensity ratioin the form (23) permits an interpretation in energy terms . Using the Fouriercoefficientsgm ([3], equations (4), (7)) we get from (23)

str Ig(t)] = IgoI 2'E,,

(24)and further

str [g(t)] =(Eg

jIgml2)/Eg=1- 1 1 1ggl 2 .

( 25)o

Eg moo

Thus, according (24), the Strehl criterion represents the ratio of the amount ofenergy coming to the imaging focus (m=0) to the total energy Eg transmittedthrough the filter-cf. [7], § 6 . It follows from (25) that strg 51 . The equalityis valid for the uniform transmissivity g(t) = 1 when go = 1 and the gm= 0, i .e . thetotal incident energy is concentrated in the imaging focus . In the case of non-uniform pupil functions the sum 11g. 1 2, m =A 0 represents the part of the trans-mitted energy coming to the further foci of the filter . This energy intensifiesthe fringes in the focal diffraction pattern due to a super-resolving pupil if comparedwith the normal Airy pattern . Therefore, keeping the value of strg as large aspossible we attain the minimum of the total energy in the fringe system .

The super-resolving pupil function that maximizes the Strehl criterion hasbeen found by Luneberg [8] in the form

1(t) = a(Av) - b(Av)J0(Ovt112 ),

(26)

where Ov is the desired two-point resolution and a, b are constants following froma normalization condition of the fixed transmitted energy through the filter . Infact, the Luneberg solution (26) maximizes the Strehl criterion (24) so that itgives the maximum of energy into imaging focus I go l 2 while the transmitted energyEg is fixed-cf. [9, 10] . However, the optimum pupil function (26) absorbsthe incident energy in dependence on the required value of Av. Therefore, thecondition of fixed transmitted energy Eg = const has led to the solution g(t) of theLuneberg problem which gives jg(t)I > 1 . This is, however, unacceptable for apassive optical system with a constant value of incident light energy . The formalnormalization used in [9] to attain jg(t)I 5 1 interferes with the original requirementEg = const.

We propose an alternative formulation of the problem discussed. Instead offixed transmitted energy Eg = const, let us require the fixed value of the focaldiffraction amplitude, i .e .

1G(v=0)=go=Yo= f g(t) dt=1 .

(27)0

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Inverse diffraction problems

583

The desired shape of the central maximum is prescribed in the following way1

G(v=0v)= f g(t) J0(Ovt 112)dt=G,

(28)0

where G is a constant . Thus, setting G=0-2 we obtain the characteristics ofthe two-point resolution AV used in [11] . The problem is to find the functiong(t) satisfying the above conditions (27) and (28) and giving a minimum value ofthe transmitted energy

1Eg = f jg(t)j 2 dt=min.

(29)0

Such a function g(t) simultaneously maximizes the Strehl criterion (24) underthe condition (27) . The sense of the proposed reformulation of the originalLuneberg problem consists mainly in that some apodization problems can then betransferred to our reformulated problem-cf . § 4. Thus, the solution of ourproblem is also applicable for apodization purposes . Naturally, having foundthe form of the solutiong(t), its final normalization to get lg(t)l < 1 is again necess-ary.

Using the standard method of variation calculus the solution of SE, under theconditions (27), (28) may be derived in the form (26). The dependence of theparameters a, b on AV is rather different from that in the Luneberg solution-cf .[9] . From (28) and (27) we get

451(AV)

( )1(OvG

w1(~A- (AV)- [01( )] 2

II~ b(w)

(AV) [ 1( )]2'

(30)11

where1

H(Ov)= I [Jo(Ovt1"2)]2dt=[J0(w)]2+[Ji(w)]2

(31)0

and (cf. [12])n

A.(Ov)=n!(2

J.(v), n=0, 1, . . . .

(32)

In terms of a, b the associated diffraction field may be expressed by means of theLommel integrals (cf., e .g., [13], p.85)

1G(v) = f l(t)J0(vt1"2 ) dt

0

Its reciprocal represents the Strehl criterion str 1(t) = 1 /E1 because of (27) .

0

(33)=aA1(v)-b (AV)22_V, [ivJo(v)J1(zv) - vJ0(Ov)J1(v)],

G(Ov) = aA1(Ov) - bH (iv) = G .

The last equality follows from (30) . The transmitted energy is equal to1

Ei= f 1l(t)I2dt=1+b2{II(Ov)-[A 1(Ov)] 2} . (34)

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The functional Eg attains for g(t)=1(t) an extreme value. Employing theSchwarz inequality (cf . [8], §50 ; [10]) we aim to show that there is no other func-tion k(t) satisfying (27) and (28) and giving Ek < El .

Let g1 (t) and g2(t) be two functions quadratically integrable . It holds thenthe inequality

1

1

1

f g1(t)g2(t) dtJ 2 _< fIg1 (t)I 2 dt f Ig2(t)I 2 dt

0

0

0

unless g1(t) and g2(t) are proportional. Applying the Schwarz inequality tog1(t)=k(t), g2(t)=1(t) we have

1I fk(t)l(t) dtJ 2 < EkEI

(35)0

if k(t) is not proportional to 1(t) . We evaluate the left-hand side of (35) . Assum-ing the validity of the conditions (27) and (28) for k(t) we have subsequently

1f k(t) [a - bJ0(Ovt1"2 )] dt = a - bG .0

But from (30) and (34) it follows that

E1 =a-bG.

(36)

Thus, for any function k(t) satisfying the conditions (27), (28) the left-handside of inequality (35) is equal to E12 . Consequently, we conclude that

El < Ek if k(t) =Al(t) .

Under the conditions (27), (28) the functional E,, has then an absolute minimumat the function (26) .

4 . The function 1(t) as an approximative solution of the apodization problemsThe light energy diffracted into the outer rings of the focal diffraction pattern

prevents us in resolving the close details of the objects differing strongly in thebrightness . Modifying the pupil function of the optical system we can reducethe intensity of the fringes . This represents a problem of apodization [13] .Two basic integral quantities have been used to characterize the measure of apodiz-ation : the encircled energy and the spreading factor ([13], p. 77) . In our notation,the encircled energy is expressed

VmI I G(v)I 2 dv2 vm

E(vm ) _ = f I G(v)I 2 dv 2/Eg ,

(37)0

f IG(v)I 2dv20

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Inverse diffraction problems

585

where we have used the Parseval theorem for the Hankel transformation (1)between the pupil function g(t) and the associated diffraction pattern G(v), i .e .

1

wEg= I Ig(t)I2 dt=2 f IG(v)l 2vdv .

(38)0

0

While the encircled energy represents the relative part of the transmitted energyEfl which is inside the circle v S vm, the spreading factor characterizes the luminousenergy diffracted outside this circle

w

~(vm)=IG(0)I2

f IG(v)1 2dv2 .Vm

Assuming further the validity of the condition (27), i .e. G(0)=1, we can write

w

I'm.(v.) = f I G(v)I 2 dv2=Eg- I IG(v)I2dv2 .

( 39)Vm

0

The apodization problems consist in finding a pupil function which eithermaximizes the encircled energy or minimizes the spreading factor . Variousnumerical approaches have been developed [9, 13, 14] . They permit one tocompute the apodization pupil function, according to the chosen criterion, withthe desired accuracy. We want to draw attention to the fact that the optimumpupil function which extremizes E(vm ) or 2(v.) differs very slightly-cf. [13],p. 99 . With these criteria we can class the dispersion factor defined as the ratio([15], [13], p. 78)

VM

I I G(v)I 2dv2vm

D(vm , vM)=vmf I G(v)I 2dv20

By the calculations of the apodizers it is most often set vM = oo and then

D(vm oo) = EgUm

I I G(v)I2 dv 2

0

(40)

The enumerated integral criteria have a common feature : providing the energy inthe circle v S v Y, is constant, i .e .

Vm

I I G(v)I 2 dv2= const,0

(42)

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their first variations are proportional to SE, :

VmSE(Vm)= -SEg~2 f l G(v)I 2 dv2 ,

S -T(vm ) = Mg ,

SD(vm, oo) = SEg/ f I G(v)J2dv 2.0

As regards the condition (42), it is practically fulfilled for the pupil functionsdisplaying the same width of the central intensity maximum Ov, i.e. satisfying(28) . Then, the central intensity maxima have the same basic sizes. As theequation SE g=0 has the unique solution (26), under the conditions (27) and(28) are satisfied, the functionals (37), (39) and (41) attain their extremum valuesfor the unique function g(t) = 1(t) assuming the validity of (43) . Thus, theoriginal super-resolving pupil 1(t) may be useful even for apodization purposes .

5 . The pupil function 1(t) in terms of the diffraction coefficientsThe problem of variation presented in § 3 may be solved in terms of the dif-

fraction coefficients y,, . The knowledge of the exact solution (24) enables us toevaluate the convergence of the pupil function expansion in terms of yn .

Let us rewrite the requirements (27), (28) and (29) in terms of yn . The firstcondition (27) means that yo = 1-cf. (2). According to (2) we can write, insteadof (28),

00G-fo(w)= I (-1)kYk/k(Ov) •

k=1

Eg= I'0

00ws l F812= 1+ ~ 1 I j rn8Ynl2 .

8=0

8=1 Ws n=1

(43)

(44)

It remains to express the transmitted energy E 0 . Using the Parseval theorem wederive from the orthogonal expansion (20) and the relation (22)

(45)

Our aim is to find the minimum of (45) under the condition (44). Restrictingourselves to a finite number of the diffraction coefficients yn, n = 1, 2, . . . N andintroducing one Lagrange multiplicator A in the form

F(-y1, Y2, . . . YN, A) = Eg + 2A [G(Ov) - G]

we transfer our problem to finding an extreme of the function F(y1, Y2, . . . y V , A) .Differentiating F with regard to all variables we obtain N+ 1 linear equations forN+ 1 unknown values of y1, Y2, . . . YN, A of the extremum (m=1, 2, . . . N)

a=L N am8

gn8)Yn+A( -1 )"1m(w)= 0,

n =1

1 ws

(46w

(-1)1/rn( v)Yn=G-/0(Ov) .

n=1

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Inverse diffraction problems

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If the system is non-homogeneous, i .e. if the point G(Ov) = G does not lie onthe normal curve /o(Ov) 0 G(Ov), the system (46) has a non-trivial solutionYh . . . , YN, A . The set {y.}, n = 1, 2 . . . . N determines by (22) the coefficientsI" j , r,, . . . P,,, . Thus, increasing the number of equations in (46) we obtain thesolution l(t) by means of (20) with the desired accuracy .

It is surprising that even only the first three terms of (45) give a fairly goodapproximation of the exact solution 1(t) . Setting Y3=Y4 = . . . = 0 the expression(45) is reduced to

Eg 1 + 12y1 2 + 720Y 2 2 .

(47)

Similarly, the relation (44) in the same approximation turns into (G=0 .2)0.2'=''/o(Ov) - Yl/l(Ov) +Y2/2(Av) .

(48)

To find the minimum of Eg in two variables yl , y 2 we shall prefer a graphical

The figure presents these circles of constant transmitted energy in the planeof yr', Y2". At the same time, they represent the curves of constant valueof the Strehl criterion for the filters with the three-term expansion (20) asstr g(t) =1 /E g . The straight lines Ov = const are constructed using the relation(48) and the transformations (49) and (50). They define the points (v2, Y2")giving a fixed value of the quantity Ov-cf . [11 ] . If we want to achieve a certainvalue of the two-point resolution, keeping the value of the Strehl criterion aslarge as possible, we must obviously advance along the curve L(y2, Y2")=0formed by the points in which the straight lines are tangent to the circles .The curve L(y1', Y2")=0 defines the desired diffraction coefficients needed forthe three-term expansion (20) of the sought pupil function 13 (t) . With the helpof (22) we get

13(t) =1+ y 11 P1(t) +

V5Y2P2(t) .

(52 )

The form of 13 (t) is presented in the inset of the figure for Ov = 2 .5, where anappropriate normalization is done to attain 1 3(1) = 1 . The straight line correspondsto the two-term approximation 12 (t) determined by v2' = 0 and y,' such thatOv(y1 ', 0) = 2 .5 . From the inset of the figure we see that str 1 2 < str 13 but thedifferences are slight except for the largest values of Ov,>„4 . The differences

construction to the analytical solution of (46) . For simplicity let us introducenew variables yl', Y2"

Yl ' = 2-,Yl/Yo = 2,rYv (49)

= (2Tr)2 Y2 1' 1 5 = 41/ 15TrY2Y2" (50 )Yo 7r

in order to obtain the curves Eg = const in the form of circles, i .e .

E,=1+

(Y1 2 + 72 12 ) .(22)2 (51)

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M. Novotny

1 2 3 , - 4

Figure 1 . (a) The characteristics Ov (the straight lines) and strg (the circles) of the filtertransmissivity g(t) in the three-term approximation . The small circle points onthe curve L(y1 ', y2 ") correspond to the Luneberg filter l(t)=a-bJo(Ovt'l2 ) givingthe minimum Strehl criterion for arbitrary value of Av . (b) The filter transmis-sivities l,,(t) in the n-term approximation which have the minimum value of theStrehl criterion for the value Ov=2 . 5 . The small circle points correspond to theexact function 1(t) .

between str 1, and str Z are negligible until AV >, 3 . 8 . Therefore l3 (t) practicallycoincides with the exact course of 1(t) calculated from (26)-the circle points inthe inset.

A sensible test offers a comparison of the diffraction coefficients statedfor the exact function l(t) and those belonging to 1 3 (t), i .e . defined by the curveL(y l ', y2")=0 in the figure (a) . Let us derive the needed expressions .

Using the recurrence properties of the Bessel functions Jm+1(v)=-vmd[v-mJm(v)]/dv and of the Bernoulli polynomials Qk_1(t)=dQk(t)/dt,cf. ([3], equation (32)), we can perform the integration of (3) where we setg(t) =a-bJo(Ovtl/2 ) and u = 0 :

IYk = (l,Qk) = f [a-bJ o (Ovt 1 I 2)]Qk(t)dt .

0

For k = 2, 3, . . . the integration by parts yields

Yk_1(AV)=(-1)kb{m=o (k 2Em)v) - 2(kl(1)) }'

(53)

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By means of the transformations (49) and (50) we construct the points yi'(w)and y2"(Ov), the circle points in the figure (a) . The values of Ov were chosenthe same as for the straight lines of constant resolution in (a) . We find an excel-lent coincidence of the tangent points of the curve L(yl ', Y 2 ") and the pointsconstructed with the help of (54) and (55) for all values of Av < 3 .4. If iv increasesabove the value 3 .4 the three-term approximation gives the values of the Strehlcriterion of l(t) with increasing error : the relative inaccuracy of str l(t) for /Xv = 4 .0is about 4 per cent .

It seems that the function 1(t), for which it has been found to have an apodizingproperty in the previous section, could surpass the known apodizers . Thetriangle points in the figure (a) are constructed for the apodizers published bydifferent authors and summarized in [13], pp . 97 and 99 . Their pupil functionsare in our notation of the form

3

T(t)=

cn(1 -t)n

(56)n=0

with different values of the coefficients cn . Evaluating the scalar products(T, Q k)=y k by repeated integration by parts we obtain the following expressionfor the diffraction coefficients Yk in terms of cn

Cn

- ncnYO- n 1+n' y' n 2(n+l)(n+2)'

1

1

y2_- I cnn [12(n+1) 2(n+2) (n+3)]'

(57)

Using (57) and the transformations (49), (50) we compute the transformed dif-fraction coefficients yi , Y 2" due to the pupil functions (56) which apodize overdifferent intervals according to either the minimum dispersion factor (41)-thefive triangle points L in figure (a), or the minimum spreading factor (39)-theremaining two points p . All the points lie under the curve L(yl ', y2") corres-ponding to the maximum Strehl criterion attained by l(t) =a-bJ° (Avtl , 2 ) forthe variable value of Av . Since, according to § 3, the pupil function l(t) repre-sents the unique solution of the problem of variation discussed in § 3, we may expectwith regard to § 4 a somewhat stronger total apodization by the Luneberg filterl(t) than in the case of the compared four-term apodizers (56) . A more detailednumerical investigation is however necessary for doing some quantitative con-clusions .

Inverse diffraction problems 589

where we used the notation (32) . Thus,

Y1(Ov)= 2{-A1(Ov)+A2(Ov)}, (54)

Y2(AV) = 1 {-Al(Av)+3A2(Ov)-2A3(w)} . (55)

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Inverse diffraction problems

ACKNOWLEDGMENTS

The author is indebted to Mrs . Z. Kucerova for her assistance in the prepara-tion of the manuscript and to Dr . J . Komrska for critical reading of the paper .

La forme due developpement en serie recemment obtenu pour la representation desfigures de diffraction a symetrie axiale nous permet de deduire la fonction pupillaire apartir des coefficients yk(u) de la serie . On donne une nouvelle interpretation du rapportdes intensites de Strehl . Il represente le rapport entre 1'energie arrivant au foyer imageet celle arrivant aux autres foyers du filtre . De plus, en utilisant l'approximation a troistermes, on trouve la transmissivite du filtre donnant la resolution desiree pour deux pointset atteignant la valeur maximum du critere de Strehl . Cette transmissivite est compareea la solution exacte proposee par Luneberg . Finalement, on montre que la transmissivitedu filtre de Luneberg represente une solution approchee des problemes d'apodisationqui utilisent les criteres integraux d'apodisation.

Die Form der kurzlich entwickelten Reihendarstellung achsialsymmetrischer Beugungs-felder erlaubt uns die Darstellung der Pupillenfunktion aus den Reihenkoeffizientenyk(u) . Es wird eine neue Deutung der Strehlschen Definitionshelligkeit gegeben. Diesestellt das Verhaltnis der Energie im Brennpunkt der Bildebene zur Energie in den anderenBrennpunkten des Filters dar. Weiterhin wird unter Benutzung der Naherung mit dreiTermen die Transmission des Filters berechnet, das die gewUnschte Doppelpunkt-Auflosung1 iefert and ein Maximum des Strehl-Krituerims erreicht and die Ergebnisse nut der exaktenLosung nach Luneberg verglichen . SchlieBlich wird gezeigt, daB das Luneberg-Filtereine allgemeine Naherungslosung der Apodisationsprobleme darstellt, die Integral-kriterien der Apodisation benutzen .

REFERENCES

MARCHAND, E. W., and WOLF, E ., 1966, J. opt. Soc. Am ., 56, 1712 .SOMMERFELD, A ., 1964, Optics (New York: Academic Press) .NOVOTNY, M., 1977, Optica Acta, 24, 551 .LUKE, Y. L., 1969, The Special Functions and their Approximations, Vol. 1 (New York :Academic Press) .

WILKINS, J . E., 1963, J. opt. Soc. Am ., 53, 420 .STREHL, K., 1895, Z. InstrumKde ., 15, 364 .NovoTNY, M., 1973, Optica Acta, 20, 217 .LuNEBERG, R. K., 1964, Mathematical Theory of Optics (University of CaliforniaPress) .

BARRAKAT, R., 1962, J. opt. Soc. Am ., 52, 264 .WILKINS, J . E., 1963, J. opt. Soc. Am ., 53, 420 .NOVOTNf, M., 1977, Optica Acta, 24, 567.JAHNKE, E., EMDE, F., and L6SCH, F., 1960, Tafeln hoherer Funktionen (Stuttgart :Teubner Verlagsgesselschaft) .

JACQUINOT, P., and ROIZEN-DOSSIER, B., 1964, Apodisation in Progress in Optics,Vol. III, edited by E. Wolf (Amsterdam: North-Holland Publishing Co.) .

LANSRAUX, G ., and BOIVIN, G ., 1961, Can . J. Phys ., 39,158.BoIVIN, A., 1964, Theorie et Calcul des Figures de Diffraction de Revolution (Paris :

Gauthier-Villars) .

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