794 J. Opt. Soc. Am./Vol. 72, No. 6/June 1982

Kramers-Kronig relations generalized:on dispersion relations for finite frequency intervals.

A spectrum-restoring filter

Rolf Hulth6n

Physics Department, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Received October 20, 1981

When the real (imaginary) part of the transfer function of a causal linear system is known for all frequencies, theimaginary (real) part may be calculated for all frequencies from the Kramers-Kronig relations. For physical sys-tems, the real part could be the refractive index or the amplitude, and the imaginary part could be the extinctioncoefficient or the phase. Experimentally these quantities are known only for limited-frequency intervals. Thispaper presents generalized Kramers-Kronig relations, from which the real and imaginary parts may be calculatedfor all frequencies from knowledge of these parts for at least partly overlapping frequency intervals. When the pro-cedure is applied to experimental data, errors are introduced. Certain types of errors of the known real and imagi-nary parts completely destroy the possibility of calculating the unknown parts, whereas others give negligible er-rors. The existence of a filter with the property of allowing the restoration of a truncated spectrum is established.The transfer function and the impulse response function of this filter are given.

1. INTRODUCTIONThe physical principle of causality transformed into frequencyspace may be expressed mathematically as a set of dispersionrelations, e.g., the Kramers-Kronig (KK) relations. Moreprecisely, for a linear system with transfer (or system) func-tion

F(w) = FR (w) + iFj (c), (1)

the KK relations state that'- 3

FR(C) =- F, c') do', (2a)7r f-X co - c

1 £ FR (C') (2b)7r FX c - co

(The bar on the integral signs means taking the principal valueand is subsequently omitted but understood when applicable.)Knowledge of FR (w) [F1 (w)] for all frequencies thus permitsthe calculation of Fj(w) [FR (w)] for any frequency. Consid-ering real systems FR (w) and F1 (w) could mean4' 5 n - 1 (n =refractive index) and the extinction coefficient k, or n2 - k2- 1 and 2nk, or the amplitude and phase of an electrical sys-tem; in the third case, the set of Eqs. (2) is generally calledBode's equations. 6 However, for technological, physical, orother reasons, the determination of FR (w) and/or F1 (w) maybe impossible below a frequency wo or above a frequency w,:optical detectors do not respond to frequencies in the far-IRor the far-UV or x-ray regions, or absorption may become toostrong to make a measurement of refractive index and/or ex-tinction coefficient possible. A KK analysis of experimentaldata therefore demands an extrapolation of KR (w) and Ki (w)into regions in which they are unknown. Such an analysistherefore is approximate.

The present paper deals with this problem, stated in thefollowing way: Suppose that F(w) is known for wo 4 |wl a C;

is it then possible to find F(w) for Iwo <w 0 and 1w1 > wl? Aunique solution, which makes the guessed extrapolation of

F(w) unnecessary, exists and is given in Section 2. A typicalsituation is the following: The refractive index n and theextinction coefficient k of a system are measured for limitedwavelength intervals. These intervals do not have to beidentical but must be partly overlapping. It is then possibleto calculate n and k for all wavelengths.

Application examples (with wo = 0) are given in Section 4.It follows that a spectrum-restoring filter exists (see Sections3 and 7).

2. MATHEMATICAL FORMULATION OF ANDSOLUTION TO THE PROBLEM

The principle of causality demands that the impulse responsefunction f (t) of a linear system be such that, when f (t) is ex-cited by x(t), the output

rX tJ (t - u)x(u)du = f(t-u)x(u)du (3)

is zero for t < 0 if x (t) is so; that is, the cause precedes the ef-fect.

Multiplying Eq. (3) by ei~t, i = w + in and integrating overall times leads to the requirement 2 -4 that

F(Q) = I f(t)ei' tdt = FR(P) + iFi(Q) (4)

be analytic for Im Q = 7) > 0, i.e, in the upper complex-fre-quency plane. In mathematical terms, the problem leadingto the KK relations [Eqs. (2)] may now be stated in the fol-lowing way: Suppose FR (w) is known, that is, FR (Q) is knownon the boundary of Im Q > 0; then find F1 (Q) such that F(Q)is analytic on Im Q > 0. This is Dirichlet's problem; the so-lution is well known 7 and yields the KK relations when Q =

w + in - w, provided that F(Q) 0 when I - [if F(Q)Fo when I Q I -X , the following is true for F(Q) - Fo].The present problem is somewhat modified: Suppose that

F(Q) is known on a region E of the boundary; then find F(Q)

0030-3941/82/060794-10$01.00 C 1982 Optical Society of America

Rolf Hulthen

Vol. 72, No. 6/June 1982/J. Opt. Soc. Am. 795

for Im Q > 0. From this F(Q), find F(w) for w $ E. The re-gion E does not have to be symmetrical around w = 0 to permita mathematical solution but will here be taken to be so fromphysical reasons 4 since FR (w) is even and F1 (w) is odd in co.Nor does E have to include all frequencies 1 I < co, but maystart with some frequency Wo, i.e., co _ Ico < WI. However,for the sake of convenience, in what follows we will take E tomean E = {c: I o I < WI). The solution to the more general casewith E = cw:o _ I c I < c w1I is given in Appendix A.

It is not possible to find a unique analytic function F(Q),Im Q > 0 from FR (co) or F1 (c), coo c Ii < co, alone. This isthe case only when wo - 0 and W1 -

Our problem is solved by the application of a theorem fromthe representation of H2 functions. From Theorem I of Ref.8 or Theorem II of Ref. 9 after due conformal mappings orfrom Theorem 1 of Ref. 10, which is quoted in Appendix Atogether with the conditions for its validity, we have for I co> co

FR X(@) = X114 cos G (c)2w7

-FR(W') sin G(w') + F1 (W') cos G(w') dw'

W -co

+ X1/4 sin G () 12W

FR (W') cos G(W') + F1 (W') sin G(w')

co'- c)

Fi.(,) = X1/4 sin G ()

27r

-FR(c') sin G(w') + F1 (c') cos G(c&)co' -c

-X114 cos G(c) 12r

X FR (w') cos G (W') + FI (') sin G (W') dfW'1_<W1 co' -

where

G(M 1 =-IIn(1+)InC Cl.4,Wr + W1

dW', (5a)

When X - -, then FRX(w) + iFix(W) - F(w) uniformly, andF(CW) is the boundary function of a function F(Q) that is ana-lytic for Im Q > 0. In short: Eqs. (5) allow F(W) to be calcu-lated for Icol > W1 from its values for I I < Wc.

3. PHYSICAL FORMULATION OF THEPROBLEM

The problem may be stated in a more physical way. Considera system consisting of two linear subsystems with transferfunctions F(W) and H(w). Since analytic functions of analyticfunctions are again analytic, the KK relations hold for thetotal system with transfer function F(w)H(w) = T(W) = TR (w)+ iTj(c); that is, Eqs. (2) give

TR (W) = FR (w)HR (W) - Fj (w)HI (w)

= 1 f T,( I) dw',7r -X W - W

Tj(w) = FR ()HI (w) + FI (w)HR ()

= - 1 J TR (W') dw'.W /- W -

Solving for FR (w) and F1 (w) gives

FR(WW)= 1 dIw'IHI2 2

WJ-_W'

HI(w) 1 TR(W')

F] =- HR((W) 1 -wTR() dw'

Here

(5b)

(6)

A detailed derivation of Eqs. (5) is given in Appendix A.Equations (5) may be written more compactly as

FR'(W) = -- C(X, w, w')dw'7r 1" @' -<I CO' @- W

+ 1 FRW) S(X, co, W')dw',7r CO'4@ '- C

FA(W) = 1J FR(W) C(X, , W')dco'71r <( 41 CO -'D

+W-Iwi W ) S(, co, o')-dco',

7r CO1@1@-

where

S(X, o, W') = - sin [G(w) -G(W'fl,2

C (X, co, WI') = -1/4 cos [GMw - G(W')].

(5a')

(5b')

(7a)

(7b)

(8a)

(8b)

(9)

HI(@) 1 p T1(W')H H12 (w + o,! - Co

IH12 = HR2(W,) + H,2(WL).

Rewriting the integrals gives

FR () =HR (W) 1 T1(Wo') dW'FRw)=HI2

Wr W d w

Hj(W) W 1 '~ TR ("') do,'IH12 7r f J <l, WIl a-@W

+HR(W) 1 X TI(w') do+H12 7r Iw'I>wi w' -aWi

H1(w) 1 TR(') w,

IHI2 W wl I CO', -CO dwt),

FM (=HR( ) 1 p TR(W') do!JHI2 7r WO' - CO

Hi(c) 1 Tj(w')IHI 2 W J 1,<wj '-w dw'

HRM() 1 f TR (W')

IHI 2 7r J., -W dow

HI(c) 1 r T1 (W')

IHI2 Wr J'I>wi W- w dw@

(8a')

(8b')

It follows from Eqs. (5) that it is possible to find a filter H(w)= HR (w) + iH1 (c) such that the two last integrals of Eqs. (8a')and (8b') vanish when A - . Identification gives a filter withtransfer function

Rolf Hulthen

X fjWj_-Wj

1X fjWj_-.j

Xf., I <W1

796 J. Opt. Soc. Am./Vol. 72, No. 6/June 1982

HP (X ) = J APe -l/4 ', Ico I ->- 2Xp- 1 /4 e-iG(co), 1W! > co,'

G(co) = 1 In (1 + X) In | J.47r Ico + c

where p is any real number. Choosing p = 0 gives

Rolf Hulthen

FRX(w) = Al/4 cos G(w) I2wr

(6) X fl-,,>,,,FR (w') sin G(w') + Fj (w') cos G (w')

W' - Wdco'

- X1/4 sin G ( I2w7

e-iG( d)IHO(X, w) -_ H(U =12X-114e-iGMl

1W < Wi

1'W >, 1

(10) XX FR(W') cos G(c') - Fj(c') sin G(co')' -co

Inserting this H(c) in Eqs. (8') also for IwI c c 1 gives

FR () = cos G (X)-

X S -FR (W') sin G(c') + Fi (w') cos G(o') d) 'x dw'<J c

+ sin G( )-

C FR(w') cos G(d) + Fj (w') sin G(cd ) (Ila)F 1w = si1 c()o

Fi(co) = sin G (co)-

Fjx(w) = -X1/4 sin G(co) 2

FR (w') sin G (w') + Fj (w') cos G (w')Wc -cW

dco'

- X1/4 cos G(w) 1

FR(w') cos G(w') - F1 (w') sin G(c')C, -c

dw'. (13b)

We close this section by observing that Eqs. (7), as a specialcase, contain the subtractive Kramers-Kronig (SKK) rela-tions' 1,12: Choosing

- cos G(W

x I

FR (w') sin G (w') + FJ (c') cos G (w')dcc

W -co1

7

FR (w') cos G G(') + Fi (G') sin G (d') dw'. (1ib)c o'- W

From what was said above it follows that Eqs. (11) are truisms(Appendix B) and contain no new information: they merelycalculate FR(W) and Fi (w) for 1Iw < wC from FR(w) and Fi(w)for I o I < w1. This implies that the application of sum rules,4

such as [from Eq. (Ila)]

FR (0) =

-FR(w') sin G (w') + Fj(w') cos G(') ,WI

gives no further information.Nevertheless, Eqs. (11) may be used to check the numerical

treatment of Eqs. (5) and are therefore useful. This will becommented on in Section 5.

The above solution for FR (c) and FJ(w) leading to Eqs. (8)is equivalent to filtering FR(w)HR(") - FJ(wo)HI(w) +i[F,(wX)Hr(wo)+ F (wC)HP (&) through a filter with transferfunction

eiG(),

M(W) = M(X, w) = -eX1(/

2

G(w) is given by Eq. (6).This filter is complementary to H(w) in

enables F(w), I I < wi, now supposing F(in this interval, to be calculated from F(w)lIeeI < wi we have from Eqs. (A7) (Appendi

HAW() = 4(W - w2) - b(w + w2),

we have from Eq. (2a)

HR( ) = 2 W2wr W2

2 - C,2

Inserting these expressions into Eqs. (7) yields

F.(W) = FISKK(W) =-FI ( OW2

2 L FR(W') dw'-- W

2 -w 2

2 )J (W22 - W'2)(C2 - w'2)

FR(CO) = FRSKK(W) = FR(W2)

+2 -(2 - 2) w W'Fj(w') dw'7r - o (w 2

2 - c' 2 )(W 2 - w'2)

Sometimes'2 the logarithm of F(w) = FR(wo) + iFI(M) =

r(w)eiP(w) is used. In such a case FR (w) is replaced by in r(w),and Fi(w) by 0(U), in the above expressions.

4. APPLICATIONS

The usefulness of Eqs. (5) has been tested in the following way:Choose a boundary function FR (w) and calculate its conjugate

I < FF (w) from Eq. (2b) [conjugate pairs FR (w) and Fj (w) are of'1w <Wi course easily found as the real and imaginary parts of the

J I >e w, (12) Fourier transform of a function f(t ) such that f (t) = 0 for t <0]. Truncate F(w) at some w = wl; then recapture F(w) forIw I > w1 from Eqs. (5). This was done for two cases (bothfulfilling the conditions' 3 of Theorem 1 of Ref. 10) first for a

i the sense that it simple case and second for a case with FR (w) and F1 (c) choseno) to be unknown to give the form of the refractive-index and extinction-coef-

w, a wi, i.e., for ficient curves in the Drude-Lorentz absorption model,4' 5 thatix A) IS,

dw', (13a)

Xfl., I >-Wl

X f1-1J>_WJ

71r fJWJ_-WJ

Vol. 72, No. 6/June 1982/J. Opt. Soc. Am. 797

I-

ID

H

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0.80

0.70

0.60

0.50

0.40

0.30

0.10

0.000 1.00 2.00 3.00

FREQUENCY4.00 5.00

Fig. 1. Known spectrum [Eqs. (14)] truncated at w = wl. By usingthe real (R) and the imaginary (I) parts for w t ) w, (dashed lines), thespectrum has been restored for w > w, (a). The solid lines show thetrue continuation of the spectrum.

X

I.-Cal

IA.03IL.

L4a.

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L.

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0.50

0.40.

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0.20.

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Fig. 2. Real and imaginary parts of the truncated spectrum are suchthat the refractive index and the extinction coefficient curves of theDrude-Lorentz absorption model 4' 5 are obtained for w : 1 [Eqs.(15)]. Graphical symbols are the same as for Fig. 1.

FR(w) = Enj 2-kj2-1 NiE- ____2_-_

i me0 (Wi2 - W2)2 + g12 w2

Fj(w) = E 2niki = NEE giwi i Meo (Wi 2 - 2 )2 +gi 2 W2

Here Ni is the electron density, gi is the damping constant,wi is the absorption resonance frequency, m is the electronmass, e is the electron charge, and eo is the vacuum perme-ability.

The first case to be tested is

1FR( ) = I +

1 +W 2

Fj (w) =c1 + W2

(14a)

(14b)(I'v c Ic = 1).

The results are shown in Fig. 1. The calculated values can bemade to fall arbitrarily close to the theoretical curves bychoosing X sufficiently large. For X = 1020, the error is lessthan 0.7% for co = 1.2(0.2)3.8 and less than 1% for X =4.0(0.2)5.

The second case to be tested is

(15a)

(15b)

which was truncated at X = 2.5. The results are shown in Fig.2. The error in FR(X) is less than 11% and in Fi(co) is less than8% in the interval X = 2.75(0.25)5 for X = 1028. Further detailsin the numerical integration will be given in Section 5.

When the present procedure is applied to a practical sit-uation, the known functions FR (c) and F1 (c) are interpolationformulas of experimental data. The problem of how to choosethese formulas when noise is present is nontrivial. 14 Theinfluence of errors, e.g., introduced by noise, will be furthercommented on in Section 6.

5. NUMERICAL TREATMENT

We are faced with the problem of numerically calculating theintegrals in Eqs. (5), the integrands of which contain clusterpoints at the upper boundary. This cluster point is removedby the substitution

x =-In\w1 + 4

leaving an infinite integration interval. Equations (5) become(transforming to a positive integration interval)

FR x(@) = K(X, c) [w sin G () (I - I2)+ cos G(co)(I3 + I4)], (16a)

Fix(c) = K(X, )[-co cos G(co)(I1 - 12)+ sin G(c)(I3 + I4)], (16b)

where

K(X,c) = -X4 wi7r W1

2+ W,2

I, = -r FR[(o'(x)] cos (yx) dfo N(co, wi, x)

f (.. .)dx,

I2= - FI[w'(x)] sin (yx)do N(w, w1 , x)

13 = X w'(x)F1 [w'(x)] cos (yx) df N(w, w 1, x)

( -'(x)FR [w'(x)] sin ('yx)14= I dx,Jo N(Cw, Wi, X)

FR = 4 - w2 +3 16 - c,(4 - W 2 )2 + c±2 (16- W2 )2 + 4W2

FI(w) = ( - + 3 2w(4 - W2) 2 ±w C2 (16 - U2 )2±+4C,2 '

- -- A\l\/// \

I // \\\. /

.

. . . . . . f

Rolf Hulthen

I WI - 2. 50

i

798 J. Opt. Soc. Am./Vol. 72, No. 6/June 1982

1.10

I .05

80.95 NA

o H=. 02 U=88. 98 A H=.01 U=7

o H=. 01 U=8* H=.01 U=10

0.85S.a0 1.00 2.00 3.00 4.00 5.00

FREQUENCY

Fig. 3. Quotient F,?calc(W)/FRtrue(w) for the spectrum in Fig. 1 forvarious values of parameters used in the numerical integration. Theaccuracy is simultaneously increased in the intervals w = (0; 1) (wherethe spectrum is known) and w = (1; 5) (where the spectrum is un-known) when one or more of the integration parameters are changed.This offers a method to check the numerical integration. H is theintegration step length and U is the upper integration boundary(approximating infinity).

N(w, c1, x) = 1 + (e-x + ex) 2(W2 + W12)W2 - W12

ay = - ln (1 + X).47r

The substitution co' = w'(x) is made by the computer program(written in FORTRAN F77LIB). The upper boundary u (ap-proximating infinity) used in the numerical integration wasreduced by subtracting the large x -value part from the inte-grand. Il (and likewise for I2, I3, and 14) thus becomes

1= - X FR [W'(X)] cos (yx) dxo N(w, cw,, x)

e - FR[w'(X)l -FR(w1) 2 --w1)2 cos (yx)dxoN(w, wi, x) 2(w2 + W12) I

+ FR(W,) S2 - 2 e-x cos (yx)dx,2(w2 + W12) fo

where

S ex cos (-yx)dx = 1/(1 + y2).

The second example in Section 4 used X = 1028, whichmeans that the integrals in Eqs. (16) are multiplied by a factorAl/4 = 107. The calculations of FR(w) and Fj (w) thereforedemand the evaluation of differences between terms of thesame value to within six or seven digits. Double-precisionarithmetic (16 decimal digits) was hence used.

Simpson's method was used in the numerical integration.The first example (Fig. 1) used A = 1020, upper boundary u =10, and step length h = 0.01; the second example (Fig. 2) usedX = 1028, u = 10, and h = 0.01. Using some other integration

method than Simpson's, e.g., Gaussian quadrature' 5 andmultiple-precision arithmetics, would increase the accuracy

in the second example (and in more-complicated cases);programs for up to 1300 decimal digits' precision are avail-able.' 6

It was mentioned in Section 3 that Eqs. (11) can be used tocheck the numerical integration of Eqs. (5) [or rather Eqs.(16)] since Eqs. (11) calculate FR(w) and Fj(w) where they areknown. Figure 3 shows the quotient FRcalc(W)/FRtrue(w) inthe interval w = (0; 5) for the first example of Section 4;FRcalc(w) has been calculated with the same set of values X,u, and h in the two intervals (0; 1) [Eqs. (11)] and (1; 5) [Eqs.(16)].

When the accuracy is increased in the interval w < 1 bychanging one or more of the parameters X, u, and h, it is alsoincreased in the interval w > 1. The deviation from the the-oretical quotient unity in the interval w < 1 is caused bycomputer limitations, by using too small a X value, and by aninsufficient numerical-integration method. The same is as-sumed to be the case for the interval w > 1. This checkingtechnique was used when calculating F(w), where 1w! > wi,in the two examples of Section 4.

6. ERROR ANALYSIS

Since all combinations of FR(w) and Fi(w), where 1w! w,are possible parts of analytic functions, the checking techniquein the preceding section cannot detect an error AF(w) in F(w)for I wI < w1. Such an error will be interpreted as the low-frequency part of an analytic function and will thus give anerror in F(w) at 1Iw > c 1 depending on AF(w), 1I! c w1, ac-cording to Eqs. (5). The situation might be the one illustratedin Fig. 4, which shows that the problem at hand is a so-calledill-conditioned problem; the solution F(co), IwI > cw1, may varyheavily with small changes in F(w), 1 coI < w1. Errors in F(w),1w!1 < w1 increasing with frequency may thus be completelydestructive to the calculated F(Cp), I wI > c,1.

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Fig. 4. A set of possible errors AFR(w) and AFI (w) in the real andimaginary parts FR (w) and Fj (co) of the known part of the spectrum(dashed lines). The error for w t1w, will be interpreted as the low-frequency parts of an analytic function that may (but does not alwayshave to) have considerable real and imaginary parts at c > co (solidlines). In such a case the restored spectrum is destroyed.

Rolf Hulthen

Vol. 72, No. 6/June 1982/J. Opt. Soc. Am. 799

An error such that FR (w) = AFR t(w) and Fz(w) = BFIt(),w I c 1, where t means true and where A and B are both close

to unity, may also be destructive. From Eqs. (5) we have

FRCalc(Wo) = AFRt(w)

+ (B -A)X1/4 I X F,(w) cos [G(w) - G(w')]dw'.27r I' @ w -

When A Fs B, the last term may reach considerable magni-tude; when A = 1.01, B = 0.99, then FRCalc(W) > 5, IcI > ci,in the second example of Section 4. Errors in F(Cw), coi < w,behaving somewhat like the real and imaginary parts of Fig.1, give errors in the region IcI > CO,, which decrease with fre-quency if the maximum of F,(w) falls within the known re-gion.

Mathematical equalizing can never make an unstableproblem more stable; this can only be achieved by adding moreinformation to the problem. Therefore knowledge of FR (w)and F1(w) in a limited interval or in one or several isolatedpoints beyond c 1 is of great value for checking the calculatedF(cw), jwI > wl. Such checking is equivalent to the applicationof a sum rule. This also implies that alternative, e.g., itera-tive,14'17 methods for solving the present problem cannot givea more stable solution. Hjowever, if the solution is reachedwith fewer arithmetic operations, the influence of computerround-off errors may be reduced, thereby making the solutionmore stable. Iterative methods may also be numerically morestable than analytic methods (such as the present one) sincethe difference between the actual values and the computedvalues in the region in which data is available can be used toupdate the estimator. The checking technique presented inSection 5 and illustrated in Fig. 3 offers a similar possibilityto control the numerical stability of the solution.

7. SPECTRUM-RESTORING FILTER

The existence of a filter with the property of permitting therestoration of a truncated spectrum of a causal system (or acausal signal) was established in Section 3. [It is a combina-tion of H(co) and M(w) that constitutes the restoration; yetwe will name H(w) as the spectrum restoring filter.] Itstransfer function H(co) = HR(w) + iHi(c) is given by Eq. (10).HR(w) is plotted in Fig. 5 for X = 104 and X- 108; HI(w) hasthe corresponding sine behavior.

The spectrum-restoring filter H(w) may be characterizedalternatively by its impulse response function h(t), which isarrived at by performing an inverse Fourier transform of H(w).We thus obtain

1 X H(w)ei-tdw27r J-

H

UI.

a:

D

LIL

11

3L

a:

2w7r Jl14U

0

-I

-I 4*

0

FREQUENCY

2FREQUENCY

Fig. 5. The restored spectrum may be obtained from the output ofa causal filter. The figure shows the real part HR (co) of the transferfunction H(w) of the spectrum-restoring filter for X = 104 and X = 108.The accuracy of the restored spectrum is increased when the pa-rameter X increases.

ho(t) = 2 Je-iG(x)e'-iX-ltdx,

GWx) = I In (I + A) In (1-)-

(17)

(6')

If we want to restore a spectrum by analysis in the time do-main, the full impulse response function h(t) has to beused.

The explicit time dependence is obtained by inserting1 8

e iXwl= E Ji(wit)e-ilarcsinx,l=-X"

where Ji(z) is a Bessel function of the first kind. By usingknown symmetries of ho(t) as well as of Jj(z), we obtain(Appendix C) for t > 0

ho(t) = hoA(t) + hos(t)

= 2hoA(t) = - E a2m+lJ2m+1(1t)7r m=0

= 2hoS(t) = : E b2nJ 2n(W 1 t),7r n=O

(18)

± -1/4

7r L @>W1

- ho(t) + h1 (t).

h1(t) tends to zero when X - -. Retaining only ho(t) andsubstituting w/w1 = x gives the approximate impulse re-sponse

where hoA(t) and hos(t) are the antisymmetric and symmetricparts of ho(t). ho(t) = 0 for t < 0. Here

(1a2m+l = - sin G(x) sin [(2m + 1) arcsin x]dx, (19a)

co

1b 2n = Jwcos G (x ) cos [2n arcsin x]dx . (19b)

\ j; A-IE8

i1-,

n ../

I

. ..

Rolf Hulthen

e -iG(-)e -iutdc

e -iG(w)e -iwtdc

800 J. Opt. Soc. Am./Vol. 72, No. 6/June 1982

Table 1. Coefficients b2S (Calculated for X = 1028) ofthe Impulse Response Function ho(t) = 4w,/7r>n=o

b2.J 2n (WI t) of the Spectrum-Restoring Filter

2n b2n 2n b2n

0 0.000005 16 -0.1679672 0.000101 18 -0.0226374 0.002395 20 0.1080056 0.022282 22 -0.0663308 0.108972 24 -0.008383

10 0.288255 26 0.05232512 0.376967 28 -0.05402014 0.122 650 30 0.029 806

These coefficients are readily calculated numerically; b2nis given in Table 1 for 2n = 0 to 30 (calculated with X = 1028).

The value of bo (50; bo 0 0 when A - o) agrees fairly wellwith the value obtained from Eq. (C3): bo z (1/2)10-7 In (1028)

= 3.224(10-6). ho(t) may now be calculated in one of twoways, either by a numerical inversion of H(w) [i.e., by usingEq. (17)] or by using Eq. (18) and the coefficients a2 m+1 and/orb2n. As a check ho(t) was calculated in both ways; the valuesobtained in the two ways agreed within 1%.

8. CONCLUSION

The present work shows that it is possible to calculate the realand imaginary parts of the transfer function of a real (i.e., acausal) linear physical system for all frequencies if both partsare known for at least partly overlapping intervals. Thismeans that, e.g., the refractive index and the extinctioncoefficient of an optical system may be calculated for allwavelengths if these quantities are known for limited wave-length regions. However, experimental data necessarilycontain errors; the frequency dependencies of these determinethe errors of the calculated real and imaginary parts. Someerror types completely destroy the calculated real and imag-inary parts, whereas others give negligible errors in these.

If the real and imaginary parts are known in one or moreisolated points (or in another interval) beyond the knowninterval, this information may be used to check the calcula-tions, since this or these values of the real and imaginary partsmust be reproduced by the present method. Errors in theknown real and imaginary parts may thus be discovered andcontrolled.

If the real and imaginary parts are known in only partlyoverlapping intervals, the common frequency parts may beused to calculate the missing part (or parts) for the remainingfrequencies (i.e., where one part is known); the known partmay then be used to check the calculations and reduce errors.A practical situation might be that the refractive index isknown for the wavelength interval X = (XA, X2) and the ex-tinction coefficient (or the absorption constant) is known forA = (XA, XA3), where X2 > X3. The extinction coefficient canthen be calculated for X = (X3, X2) by using the present methodin the interval X = (Xi, X3) and by using the refractive indexin the interval X = (X3, X2) as a check.

APPENDIX A. DERIVING EQUATIONS (5)

Quoting from Theorem 1 of Ref. 10 (correcting a misprint),we define analytic functions on the upper half plane Im(z) =Im(x + iy) > Oby

gx(z) = \hx(z)1 hS x~* (tg(t) dt,211ri fE t- z

h^(z) = - 1 (1+ X) E l+tz 1hz)=exp I~n (1+ ) z d1t2

(Al)

(A2)

Equations (Al) and (A2) both hold for Im(z) > 0. Theseequations, contrary to what is stated in Ref. 10, are here notsaid to hold for Im(z) = 0, i.e., on the real axis; this is wherewe want to evaluate gA (z). This will be done by letting z =x + iy - x while y > 0.

hx* is the complex conjugate of hx, and E is a set of positivemeasure of the real axis. The theorem states that if f e H 2(R)and g = fon E, then, as X -@ , gx -O f uniformly on compactsubsets of the upper half-plane, and 9X - fAl2 - 0. As anintegration interval we choose EA = [t: xo < IRe(t)I = I|I <xi, Im(t) = =Yi >0] 0- E = (t: t =r, xo < xi) whenYi - 0. The integral in Eq. (A2) is

1+z d-dt 1 t dt,EA t-z 1+t 2 JEA t-z 2 SEA 1 + t2

where

S dt d(r + ia)JEA t -Z JEA (T-X) -i(y -O)

=Iln XO+XXI- XIXo- X X1 + X I

+ i-rR(x, xo, x1)

when y - +O and -- +O while y - a> 0.

R(XX0, gxi) = (1 for xo < lx I <xi10 elsewhere

Further,

2t dt I (I+t 2 )-2riJEA 1 + t2 EA

when a - +0; this last contribution comes from encircling thepole t = +i.'

We may now go down to the real axis and obtain

hA(x) =exp 1--In (1 + X)( 47ri

X {ir[l + R(x, x0, xi)] + In Xo + X Xi - xI - x xi + X

= (1 + X)-[l+R(xxox1)]/4

Xexp 1(1+ IX o-xxi-xH't4r ht-i Xs + X

that is,

hx(x) = (1 + X)i1/2

eiG(x) for xo < lxi I xi,

hx(x) = (1 + X)-i/4eiG(x) elsewhere,

G(x)= ln(1+X)ln Xo+xix47r xo-x Xi + X

(A3a)

(A3b)

(A4)

Rolf Hulthen

Vol. 72, No. 6/June 1982/J. Opt. Soc. Am. 801

We further have

EA h(*(t) dt = hx* h(t)g(t) dtfEA t -Z EA (Tr-X) - i(y - a-)

r hA*(t)g(t) dt + ir (yI-)h*(t)g(t)EA T-X d I JEA (T-X)2 + (y - U)2

X hx*(-)g(T) dT + iw-h\*(x)g(x)R(x, xo, xi),XOS IT I <X1 TX

(A5)

wheny,oa -Owhiley-a >O. Forx 0o <xi jX 1,wethereforehave (X >> 1)

g^(x) = eiG(x) 1 S e-iG(T)g(r) dr + 1g(x).2Tri Xo ITI xI T-x 2

(A6a)

eiG(x) - 1 when xo - 0 and x1 i o. Equation (A6a) thengives, after separating out real and imaginary parts, the KKrelations [Eqs. (2)]. This is not the case if we use Eqs. (Al)and (A2) directly for Im(z) = y = 0.

For Ix I < xo and Ixi > xi we have (X >> 1) from Eqs. (Al),(A3), and (A5)

gx(x) = X1/4eiG(x) 1 f e-iG(&)g(T) dr, (A6b)27ri Xo<ITIxl -T-X

Again separating out real and imaginary parts and changingnames of variables and functions (g - F, h - H,r -T w', x -

W) to correspond to physical conventions gives

FRX(CO) = 1/4 Cos G(c) -2~r

o f0<W'I-<Wi-FR(w') sin G(w') + F1 (w') cos G(w')

Ci) - W

+ X1/4 sin G(w) 1

x J FR(w') cos G(co') + F1 (w') sin G(Wc')d(a, - W

(A7a)

APPENDIX B: PROVING THAT EQUATIONS(11) ARE TRUISMS

Define

'R(R ) = cos G( )-

-FR(c') sin G(c') + F1(c') cos G(c&) dw'W, -W

+ sin G(c)-

FR(c') cos G(c') + F,(w') sin G(W')W, -c

7r f1z'114

1

1+-T' .Jj'I4

FR(G') sin [G(c) - G(G')] do'1 cos - G(

Fi(c') cos [Go ()- GG01") do'. (B 1)CO' - coW1

Here

G( ) - G(o') = G(wix) - G(cwly)

= ly [ln ( X)- In 11Y)

where

'y =-ln (1 + X),4ir

x =W/Co1, jxj <1,

Y = '/°i, Iyj < 1

G(wix) - G(wty) = 2y[(y - x) + 1/3(y 3 - X3 )

+ 1/5(y5 - X5 ) + . . .1]

(B2)-, [e(y - x) + u (x, y)],

where we have used the series expansion

ln( 2) = -2x+3x3+x5+...)

holding for Ix I <1.

* *tFR(w)

'PR (WlX )

1 ,r FR(wOy) sin [2y(y - x) + 2yu(x, y)] d7r J- y-x

Fi"(c) = X1/4 sin G M 22~r

XSwod(I<Wi-FR(d') sin G(w') + F,(w') cos G(w') d

c) - W

- X1/4 cos G() 127r

X FR( ') cos G(d') + Fj(c') sin G(w') d lf-o0 11d1 <Wj @t-

(A7b)

from which Eqs. (5) follow when Co = 0 and Eq. (6) followsfrom Eq. (A4).

+ 1 f I1 Fj(wly) cos [2-y(y - x) + 2-yu(x, Y)] d-1 y -x

=1 J FR(w1y) cos [2yu(x, y sin 2y(y - x)yw J-i y- x

+1 J1 FR (wly) sin [2-yu(x,y)] cos 2y(y - x) dy

+- Fj(wy) cos [2yu(x y)] cos 2,(Jy - x) dy7r -1 y-x

-- f( FI(C1y) sin [2yu(x, y)] sin 2'y'(y - x) dy.7r -1 y-x

(B3)

Rolf Hulthen

X fjWj--(01

X fwl--Wl

802 J. Opt. Soc. Am./Vol. 72, No. 6/June 1982

But

lim 1 sin 2zy(y -x) = 5(y-X).r-.0 7r y-x

and u(x, x) = 0 from Eq. (B2).

1 r' in 2,y(y - x)J FR(w1y) cos [2-yu(x, y)] s dygr -1y-x

= FR(wlx) cos [2yu(x, x)] = FR(wlx) = FR(w), (B4)

-J Fj(wiy) sin [2yu(x, y)] sin 2 y(y - x) dyW f-1 y-x

= F1(wcx) sin [2yu(x, x)] = 0. (B5)

Further,

S- FR(wiy)sin[2yu(xy)] cos 2-y(y - x) d

-X ( . )dy + f1 (...)dy

and

Jf FR (w1y) sin [2yu(x, y)] cos 2-y(y - x) d-1 y-x

S-ln2y(1+x) 1 - 21 ,'j

X sin 2yu tx, x - 2 et)] cos (e-t)dt

- -FR (w1 x) sin [2yu(x, x)] J cos (e-t)dt = 0

when y X; here the Substitution y = x - et2y has beendone. Analogously,

1FR (w ly) sin [2yu(x, y)] cos 2y(y - x) dy - 0

when y -

lim f FR(wly) sin [2-yu(x,y)] cos 2 y(y-x)dy =.Y sJo-1 y -x(B6)

Further,

1 F1(w1y) cos [2yu(x, y)] cos 2y(y - x) dy-1 y-x--X (.. .)dy + (...)dy,

where

JE (...)dy= fXFI(wy) cos [2yu(x, y)]

cos 2y(y - x)y-x

=-- Y(i+X) FI [i (X - e)]X cos [2-yu(x, x - e-t)1 cos (e-t)dt

2Fc

-FI(wix) E.cos(e-t)dt

when y -p o. Analogously,

1Fj(wiy) cos [2yu(x, y)] cos 2(y - x) dy

+Fi(wix) cos (e-t)dt when y -y

lim JR Fj(coly) cos [2,yu(x,y)] Cos (Y ) dy = 0.t. t -1y -x

(B7)

Inserting Eqs. (B4)-(B7) into Eq. (B3) gives 1?R (w1X) = FR (O),whereby Eq. (Ila) is shown to be a truism. Equation (lib)is treated in the same way.

APPENDIX C. TIME DEPENDENCE OFIMPULSE RESPONSE FUNCTION h(t)j

From Section 7 we have

Inserting 18

(17)

(6')

ho(t) = ' J e-iG(x)e-ixw'tdx,27r -E

G~x) = I In (1 + )X) In 1 )-xi4ir\=+xJ

e-ixwjt = Z Jj(wit~e-iiarcsinx

gives (reversing the integration and summation order)

ho(t) =1 wl e-iG(x)e-iarcsinxdx] Ji(wit)27r 1=- -

But

f e-iG(x)e-iiarcsinxdx

= -2 Jr sin G(x) sin (1 arcsin x)dx

1+ 2 Jo cos G(x) cos (1 arcsin x)dx,

2(al + b1)

since G (x) and aresin x are both odd in x.It follows from Eq. (C2) that

a-1 = -al,

b- 1 = +bj,

aO = 0,

whereas

bo = J cos G(x)dx = 2 J cos (-yt)dt

=-2 f d( ) coss(t= ylr[sinh(-7r)]1 2,yre-71

X-1/4l nWk

2 .

for large values of X, i.e., bo - 0 when X -.

(C2)

Rolf Hulthen

Vol. 72, No. 6/June 1982/J. Opt. Soc. Am. 803

We have used 19

Swsin (eyx) dx = I - 7 [sin h(-y)]-1.o 1 +ex 2, 2

Further,'8

J-1(z) = (-101W(Z,

which together with Eq. (C2) inserted into Eq. (Cl) give

ho(t) = j2 (a-i + b-j)J-j(wit) + (al + bj)Jj(wit)7X 1=1

= E [1 - (-1)1aiJi(wit) + [1 + (-1)1]bjJj(cot)7r 1=1

=2 1 i a 2 m+1J2 m+1(W1t) + 2 1 j b2nJ2.(W10,7r m=O 7r n=1

where

a2m+1 = - 4' sin G(x) sin [(2m + 1) arcsin x]dx,

1b2n = f cos G(x) cos [2n arcsin xIdx

from Eq. (C2). Further,' 8

Jj(-Z) = (-1)1j1(z),

that is, ho(t) may be written as

ho(t) = hoA(t) + hoS(t),

where hoA (t) is odd (antisymmetric) and(symmetric) in t, and

hoA(t) = 2- " a2m+lJ2m+1(wlt),7 m=O

hos(t) = 2 W' j b2nJ2n(W1t).1r n=1

For t > 0 we thus have2 0'2'

ho(t) = 2hoA(t) = 2hoS(t)

= 4- w a2m+lJ2m+1(W1t)7r m=O

= 4 1 j b 2 nJ 2 n (Wit).7r n=l

(19a)

(19b)

hoS(t) is even

(18)

For t < 0, ho(t) = 0, i.e., hoA(t) = -hoS(t); this was checkednumerically and gave I [hoA(t) + hoS(t)]/[hoA(_t) + hos(-t)] I< 10-5, when hOA and hos were computed separately for t <0. The coefficients b 2n are given in Table 1 for X = 1028.

ACKNOWLEDGMENTS

Parts of this work were presented at the Ninth Nordic Semi-conductor Meeting, June 1980, Sigtuna, Sweden.

The author wishes to express his gratitude to J. Anianssonfor clarifying discussions on Ref. 8 and to L. Huldt and N. G.Nilsson for reading the manuscript and making several con-structive suggestions.

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Rolf Hulthen