Free-space propagation model for coherence-separablebroadband optical fields

Stanley R. RobinsonDepartment of Electrical Engineering, Air Force Institute of Technology, Wright-Patterson AFB, Ohio 45433

Paul S. IdellRome Air Development Center (OCSE), Griffiss AFB, New York 13441

(Received 14 July 1979)

A free-space propagation model for broadband (spectral purity A/4A - 1) optical fields is developedbased on a Karhunen-Lo6ve (KL) expansion of the time-varying portion of a coherence-separablebroadband optical envelope. For long characterization intervals it is found that the eigenfunctions ofthe KL expansion are approximated by complex exponentials of a Fourier-series expansion; thecorresponding eigenvalues are approximated by samples of the temporal power spectrum, sampled atthe harmonic frequencies of the Fourier-series expansion. The resulting modal expansion provides anintuitively simple interpretation of the propagation of broadband fields and allows the output fieldstatistics to be calculated easily.

INTRODUCTION

The propagation of monochromatic light waves through freespace may be described using the Huygens-Fresnel principle.'For suitably narrowband (i.e., quasimonochromatic) opticalfields and restricted, but useful, geometrical situations, thepropagation of light has been described using the Huygens-Fresnel integral.2 The restriction on temporal bandwidth fornarrowband fields is that the bandwidth of the time flucta-tions of the field Af is small compared with the center opticalfrequency Jo

AfIf o << 1,

or, equivalently, the spectral purity,

Xo/AX >> 1, (1)

where AX is the linewidth of the optical radiation, Xo is thecenter wavelength, Xofo = c, and c is the speed of light in avacuum.

In this paper we present a free-space propagation model forbroadband optical fields based on a modal expansion of thetime-varying portion of the electric field envelope and theHuygens-Fresnel integral. By broadband it is meant that thespectral purity Xo/AX is on the order of unity, clearly not

satisfying the narrowband condition (1). For comparison, thespectral purity of several representative broadband andquasimonochromatic sources are listed in Table I.

The propagation model assumes that the spatial variationsof the input field are independent of time so that the space-time correlation function can be written as the product of thespatial and temporal correlation functions. This coherenceseparability allows the source field to be represented simply,lends tractability to the mathematics, and provides an intui-tively simple description for the propagation of broadbandfields. Furthermore, the model allows the second-momentstatistics of the propagated field to be computed easily.

In Sec. I the optical envelope of a broadband optical fieldis defined, and a Karhunen-Loeve (KL) expansion of thetime-varying part of the optical envelope is developed. Sincethe coefficients of a KL expansion are uncorrelated, it is shownthat the source field may be expressed as the sum of temporalmodes that are pairwise uncorrelated. This uncorrelatedproperty of the expansion coefficients allows each temporalmode of the broadband field to be propagated individually,providing the basis for the broadband propagation modelintroduced in Sec. II. By using the propagation model, thespatial correlation (second statistical moment) of a broadband

432 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 432

TABLE I. Spectral purity of typical optical sources.

Linewidth Spectral purtiySource (AX) (W/AX)

Highly stable gas laser 3 1o-7 A 5 X 1010 NarrowbandPoor quality solid-state 10-2 A 8 X 105

laser3

White light 3000 A 1.8(0.4-0.7,um)

ir Window 2 am 2 Broadband(3-5 ttm)(8-12 gm) 4 jm 2.5

field propagated a distance z from the source is calculated.The idea of "coherence length" in the output plane is dis-cussed, and the broadband model is shown to yield well-knownresults for the case of narrowband sources.

1. MODAL DECOMPOSITION FOR BROADBANDSOURCE FIELDS

Consider the free-space propagation channel, pictured inFig. 1, consisting of two parallel planes P1 and P 2 separatedby a distance z. Plane Pi is located at z = z1 containing pointsri = (x1,y1 ); plane P2 is located at Z2 = z1 + z and containspoints r 2 = (x 2,y2). Let U 1(rl,t) represent the complex op-tical field envelope of a temporally broadband source fieldpropagating along the positive z axis (defined in plane P1 ). Ifthe temporal fluctuations of the field are centered at opticalfrequency fo, the scalar electric field fluctuations of the elec-tromagnetic field may be written1

E1 (r1 ,t) = Re{U1 (r1 ,t) exp[-j27rfot]b, (2)

where El(rl,t) is the normalized scalar electric field given inV m-

1 Q-112 and Re {-} means "the real part of."

We suppose that the spatial variations of U1(r 1,t) are in-dependent of time, so that the field is coherence-separable andmay be represented by

Ui(r 1 ,t) = ui(rjw(t),

where u 1(rl) represents the spatial part of the input field andw(t) is the broadband, time-varying portion of the field.

Strictly speaking, this coherence model is applicable to sourcefields whose spatial variations are constant for all time. Thisrestriction can be loosened somewhat for the case where thespatial variations are constant or slowly varying over timeintervals comparable to the characterization interval T of themeasurement.

The spatial and temporal parts of the optical field u1(rl)and w(t) are assumed to be complex, zero-mean randomprocesses whose real and imaginary parts are uncorrelated andidentically distributed. Their second moments may bewritten

(ui(ri)u*(r'1)) = Ri(rirj),

(w(t)w*(t')) = R,,(tt'),

(3)

(4)

where (-) denotes ensemble averaging. Furthermore, if weassume the statistics of w(t) are stationary, the correlation ofthe temporal fluctuations of the field may be written

(5)

(6)

Rw(t, t) = Rw(t- 0 = Rw(T),

R. (,T) = F- 1 {$w (f) ,

where r = t - t', Ff I H-} denotes the inverse Fourier transformwith respect to f, and Sw (f) is twice the power spectrum ofeither the real or imaginary part of w(t). For typical white-light applications, the spectral content of w(t) may be con-sidered to range between 0.4 and 0.7 gim, corresponding to atemporal bandwidth Af equal to 3 X 1014 Hz. For conve-nience, the power spectrum S,.(f) will be normalized, sothat

S f =SW(f)df=i (7

Thus, all the power in the complex field is assigned to thespatially varying portion of the field envelope u1(r1 ).

Consider the following modal expansion of the complexrandom process w(t) along a complete orthonormal set(CON) of basis functions $on(t)I over a finite time interval[-T/2,T/2]:

w(t) = l.i.m. N wn0n-t)N-.- n=-N

where

for- < t <-2 2

_ T/2

(8)

(9)

L.i.m. denotes limit in the mean, implying a mean-squareconvergence of the sum (8), and n is the integer index of thenth temporal mode. Note that 10n(t)) is a set of complexfunctions yet to be specified. Also, since w(t) is assumed tobe a zero-mean, complex random process, the {wn}I are zero-mean, complex random variables.

By proper selection of basis functions I n(t)I it is possibleto expand w(t) so that/the coefficients of the expansion 1wnIare pairwise uncorrelated:

(Wn~Wn) = (Wn) (Wn') = 0,

O U2(tru2Ft)

Output Field

FIG. 1. Free-space propagation geometry.

433 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

(10)

for n it n'. (The second equality occurs because the coeffi-cients are zero-mean random variables.) A necessary andsufficient condition for the fwn I to be uncorrelated is that thebasis functions 10n(t)j are solutions to the Fredholm equa-

Stanley R. Robinson and Paul S. Idell 433

Input P.ane: P1

Yl

r.

X1

U1(r1,t) KInput Field

Plane P2

t Y'2

X2z

(7)

I

tion4

T/2-Y.t~ (t) = 2 R. (t,00')(Ct) dt',

.1-T/2

for -T/2 < t < T/2, (11)

where eyj are the real eigenvalues associated with theeigenfunctions {0n(t)I. The expansion of w(t) on a CON setof eigenfunctions over an interval yielding uncorrelatedcoefficients is known as a Karhunen-Loeve (KL) expansion.If the basis functions 10n(t)1 of Eq. (8) are solutions to (11)then the modal expansion of w(t) is such an expansion.

Results from linear integral equation theory5 state that anysquare integrable kernel R,, (t,t') of (11) may be expanded ina series

R.(t,t') = E Znfn(00*nW)n

T Tfor-2 < tt' < 2 v (12)

where the convergence is uniform for - T/2 < t,t' < T/2.Equation (12) is called Mercer's theorem. It can be shown6

that if the correlation function of a zero-mean, complex ran-dom process w(t) can be expanded in a form (12), the modalexpansion given in (8) will converge in the mean square.Thus, for any correlation Rw(t,t') that is continuous andbounded for -T/2 < t,t' < T/2, the modal expansion of w(t)given in Eq. (8) will converge in the mean square to the processw(t).

For stationary random processes characterized over longtime intervals [-T/2,T/21, it can be shown4 that the eigen-functions 1kin(t)l and associated eigenvalues ln, which aresolutions to the Fredholm equation (11) can be approximatedby7

T Tfor-- < t < T (13)

2 2

and

PYn = (1IT)Sw(n/T), (14)

where T is the characterization interval in seconds andSw (nIT) is the power spectrum of the complex random processw(t), defined in Eq. (6), sampled at frequencies (n/T) Hz.Since the expansion coefficients twnI are by definition pairwiseuncorrelated, it can be shown that

(wnwn,) = (1/T)Sw(nIT)6nn', (15)

where rnn' is a Kronecker delta.

Therefore, for long characterization intervals, the eigen-functions of the modal expansion (8) have been shown to beapproximated by complex exponentials of a Fourier-seriesexpansion over time interval T, and the corresponding ei-genvalues Ihnj are approximated by samples of the powerspectrum S,. (f) sampled at the harmonic frequencies of theFourier-series expansion. The magnitude of T needed for thevalidity of the approximation depends on how quickly Sw (f )changes near frequency f = n/T. For smooth spectra, longT means long compared with the reciprocal bandwidth of thefluctuations of the optical envelope:

T >> 1/Af. (16)

For white-light applications the bandwidth Af 3.0 X 1014Hz. Therefore, for a characterization time T much greater

than 6.7 X 10-13 s, the approximations stated in (13) and (14)are valid.

The importance of the modal expansion for the complexfield envelope given in Eq. (8) is the following: Since Max-well's equations governing propagation of electromagneticwaves for linear media are linear in space and time, each termin the expansion-or each temporal mode-propagates in-dividually. In terms of optical fields this means that the nthtemporal mode of the broadband field envelope may bepropagated as a monchromatic wave at frequency fn = fo- n/T. By superposition, the output broadband field issimply the sum of all the individually propagated modes. InSec. II the superposition of individually propagated mono-chromatic modes will be used to investigate the coherenceproperties of broadband optical fields.

As an alternative approach to obtain a modal expansion forthe field envelope, one might expand w(t) in a Fourier seriesover an interval T. It is well known 8' 9 that the coefficientsof such an expansion become uncorrelated as the expansioninterval gets long. For band-limited power spectra Sw (f),solutions to Eq. (11) are recognized as prolate-spheroidal wavefunctions.10 It is interesting to note that for this case, pro-late-spheroidal wave functions are approximated by complexexponentials when the characterization interval becomeslong.

II. COHERENCE PROPERTIES OF BROADBANDOPTICAL FIELDS

The propagation of monochromatic fields from input planeP1 to output plane P2 is governed by the Huygens-Fresnelintegral2 which may be written

u2(r2 ) = e-z J'ul(rl)exp [j2 I r2u-r12] drl, (16)

where ui(r1 ) is the complex field amplitude of the input field,u2 (r2 ) is the field amplitude of the output field, and X0 is thewavelength of the light. In Eq. (16) the paraxial (Fresnel)approximation has been used. For many applications, theoptical fields of interest are confined to a region about the zaxis whose maximum linear dimension is small compared withthe propagation distance z. For these situations, the Fresnelapproximation to the Huygens-Fresnel integral is very good.This form also allows the computations associated with wavepropagation and diffraction to be greatly simplified, allowinga "systems" type representation of these effects.

By using the modal expansion for a temporally broadbandoptical field given in Eq. (8), the output field mode U2 n(r 2,t)due to a monchromatic temporal mode at optical frequencyA = c/Xn = fo - n/T at the input plane can be written

ejknz o[kn 12U2n(r2,t) = jp FUln(rl,t) exp Ljn|r2 - rl2 dr,

jAnZ f12z I

-T <t<Tfor-<t< (17)

where

pn = Xot-1(n),

An = 27/Xn = (2r/Xo)t(n),

t(n) = 1 - n/foT,

(18)

(19)

(20)

434 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980

0,,(t) = exp[+j2r(n/T)t]

Stanley R. Robinson and Paul S. Idell 434

and U1,(rl,t) is the nth temporal mode of the broadband-fieldenvelope Uj(r,,t) located at plane PI:

Uln(rl,t) = ul(rl)wn exp[+j27r(n/T)t]. (21)

Note that both input and output field modes are basebandrepresentations of the nth temporal mode at the input andoutput planes, respectively, taken with respect to an opticalcarrier at center frequency fo = c/X0.

For broadband (white-light) fields, the output field U2(r 2,t)can be written as the sum of the individually propagatedtemporal modes

U 2 (r 2,t) = E U 2 n(r 2,t)T T

for-- < t < -, (22)2 2

where U2n (r 2,t) is given in Eq. (17) and the sum converges inthe mean. By using Eq. (22), the correlation of the broadbandoutput field in plane P 2 can be written

F2 (r 2,r 2 ,t,t') = (U 2(r 2,t)0U(r'2 ,t'))

= E (nz )-2 dr J'drilln(ri,r,t - t')n f fX expUj(kn/2z)(Jr 2 - r112 - Ir' - r&)]

T Tfor - T < -<

2 2'

(23)

(24)

where

rFn(r, 4r, r) = (1/T)S. (n/T)Ri(r,, r'l) exp[U27r(n/T)T]

(25)

is the nth temporal mode of the correlation of the source fieldgiven in Eq. (21). The output field correlation of r 2(r 2,r2,t,t'),or the mutual correlation function, expresses the mutual co-herence of light fluctuations at points r 2 and r2' in the outputplane, where the fluctuations at r2 are taken at time t, andthose at r 2' are taken at time t'.

The spatial correlation of the output field is calculated fromEq. (24) when t -t' = T = 0:

r 2 (r 2,r 2) A r 2(r2,r2 , r = 0) (26)

= (XnZ)- 2 Jdr, fdr'rln(ri,rD)n f f

X expU,(kn/2z)(Jr2 - ri12 - r2- ra12)]

T Tfor-- < t < -, (27)

2 2

where

the spatial correlation of the output field may be written fromEq. (27),

F2(r2 ,r2 ) coherent = E (XnZ ) 2(1/T)S.(n/T)source n

X f driu1(r1)expU(kn/2z)1r 2 - r112 ]

X f dr'iu (r')exp[-j(kn/2z) I r2 - r 12]

T Tfor-- < t < - . (30)2 2

For broadband, spatially incoherent sources Eq. (27) be-comes

F2(r2 ,r'2 ) h = TG(Xnz)-2(1/T)Sw(nIT)

source

X eVIn 5 drIi(ri)exp[-j(kn/z)ri-(r 2 -r)]

T Tfor-- < t < -, (31)

2 2

where

4'n = (kn/2z)(1r 2 12 - Ir212)

and

Ii(r,) = ui(r1)u*(r,)

is defined to be the intensity of the source in W m-2 . Notethat Eq. (31) is an extension of the Van Cittert-Zernicketheorem' for temporally broadband sources; that is, for eachtemporal mode n the mutual intensity of the light at theoutput plane is just as prescribed by the Van Cittert-Zernicketheorem. The spatial correlation for white-light, extendedsources (31) is, therefore, the sum of all the mutual intensitiesfor each temporal mode. Also, since the spatial variations ofthe source field are assumed to be wavelength independent,the contribution from each temporal mode is weighted by thetemporal power spectrum So (f) evaluated at the appropriateharmonic frequency.

If we bring the sum over temporal modes inside the integralsover r, and write

kn = ko + kon/foT,

then we can rewrite Eq. (27) in the following form:

F2 (r2 ,r 2 ) = (oZ)-2 Jadri fdrtRi(riri)

X AX1(/2z0)(1r2 - r112- lr2- r&)]rln(rir ) A rln(r,,r', T = 0)

= (1/T)Sw(n/T)Ri(ri,r').

(28)

(29)

The output spatial correlation function r 2(r2 ,r2 ) has also beencalled the mutual intensity' of the output field. It expressesthe spatial coherence of light at two points r 2 and r 2 in theoutput plane when both points are considered at the sametime. For white-light sources Eq. (27) is directly related tothe fringe visibility of a Michelson stellar interferometerl forextended, white-light sources with arbitrary coherenceproperties.

For spatially coherent but temporally broadband sources,

X expU(ko/2z)(Ir 2 - r 2 - 1r4- r12)],

T Tfor--<t<-

2 2

where

fW (r) = (1IT) E 42(n)S&(n/T) exp[-j2ii-T(n/T)].

(32)

(33)

For wide process bandwidth Af and long characterization timeT, the sum over temporal modes may be approximated by an

435 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 Stanley R. Robinson and Paul S. Idell 435

Al A° V (Hz)2 0 2 f0

FIG. 2. Relationship between the quadratic term 42(v) and the temporalpower spectrum SW(v)

integral over v

fi.(-) C- f2(v)S.(v) exp[-j27rrvj]dv (34)

= F0 14 2(v)S.(U)} (35)

where t(v) = (1 - v/fo) and v has been substituted for n/T asthe integration variable.

Equation (32) indicates that the spatial correlation of abroadband field can be calculated from a twofold spatial in-tegral over (i) the spatial correlation of the source R,(r ,,r)and (ii) the contribution from differential path delays of allthe temporal modes, written as Rw(.). As shown in Eq. (35)the differential path contribution to the spatial correlationcan be approximated by the Fourier transform of a weightedpower spectrum. The relationship of the quadratic term 42(v)to a typical power spectrum is shown in Fig. 2.

Expanding

02(v) = 1 - (2v/fo) + (V/fo)2

and using the differentiation theorem for Fourier trans-forms,' 2 we may write t,, (r) in terms of the first two deriva-tives of the temporal correlation function R,, (r)

JOr 1 02W(-) = R,, (T) + J -T R,,,(-) - 2 R.2 W(T), (36)

irfo (z (2rrfo)2 Or-2

where the correlation of the stationary, wideband process w (t)is given by Eq. (6).

For any wideband power spectrum of practical interest,R, (r) tends to drop off in magnitude for increasing argument.Indeed, if the optical envelope is narrowband,

nIT << fo

for all temporal modes with significant energy, then 42(V)1 and

R. (-r) Rw TR.

To first order then for broadband optical fields, this tendencywill cause a "windowing" effect in integral (32). In terms ofspatial coherence, there exist regions in Pi (consisting ofpoints r1 and r4) and P2 (consisting of points r 2 and r4) forwhich the spatial correlation function r 2(r 2 ,r2 ) is negligible.If the spatial correlation of the source is stationary for allseparations Iri -ri2 in the source plane, the output coherencer2(r 2 ,r4) is also stationary and may be written r 2 ( r 2 - 41).The separation Pc = -r24-1r for which the output correlationfunction has significant value is called the coherence dis-tance.

Let the spatial correlation of the source be written

Rj(rj,r;) = ,j(pj)uj(rj)u*(rj), (37)

where

Pi = Irl -4r

and hi(Pi) is defined to be the complex degree of coherenceof the source, such that spatially coherent sources are definedas those for

A,(r,,rl) = 1

and spatially incoherent sources are defined so that

Aj(rjrl) = 6(r, - r).

(38)

The spatial correlation of the coherence-separable fieldU 2(r 2,t) can then most generally be written

P2 (r2 ,rD) = (X\oZ) S dr, 5dr',4(pj)uj(rj)ul(r4)X & VAl/2zc)(Ir2 - r,12 - Ir2 - r412)]

X expUj(ko/2z)(jr 2 - r12 - r4 - r2)]

T Tfor---<t <

2 2(40)

Special cases of the output spatial correlation function forsource fields having various combinations of emission spectraand spatial coherence derived from Eq. (40) are presented inTable II. Monochromatic fields are assumed to have im-

TABLE II. Special forms of output special correlation for coherence-separable source fields.

Emission Complex degree EquationType of source spectrum of coherence Output spatial correlation P2 (r2 ,r 2 ) number

Monochromatic, (1/koz)2 S drul(r,) expU(ho/2z)1r 2 - r,12 1spatially coherent 2 3(f) 1 X f drlu4(rl) expl-j(ko/2z)Ir 2 - r4j2] (41)

Monochromatic,spatially incoherent' 5(f) B(r,- r) (1/Xcz) 2 [exp[U4o]

X X drll(r1 ) exp[-j(ko/z)rl-(r 2 - r')] (42)

Broadband, (1/XOZ)2 f dr1 f drl uj(rul)u(rl)

spatially coherent S.(f) 1 X Rw[(/2zc)(1r 2 - r1 2 - 1r - r12)1

[Eq. (41)] X exp[j(ko/2z)(jr 2 - ru12 - 14r - r'fl2 )] (43)

Broadband, (1/Aoz)2 expUi4oJspatially incoherent S.(f) 5(r1 - r) S drjIj(rj)exp[-j(k0/z)rj-(r 2 - r)][Eq. (42)] RwI(1/2zc)[Ir 212 - r112 - rl.(r2 - r2)]I (44)

436 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 Stanley R. Robinson and Paul S. Well

pulsive emission spectra given by Su(f) = 6(f), and the spatialcoherence of coherent and incoherent sources are modeled byEqs. (38) and (39), respectively. Note that Eq. (42) is iden-tical to the Van Cittert-Zernicke result for quasimonochro-matic, spatially incoherent sources, and to Eq. (31) with n= 0. Equations (43) and (44) are identical to (30) and (31),respectively. In (43) and (44) the sum over temporal modeshas been brought inside the spatial integration, and incor-porated into the definition of Rw(.). Note also that in thelimit as S, (f) becomes impulsive, Eq. (43) is the same as (41)for spatially coherent source fields, and Eq. (44) is the sameas (42) for spatially incoherent source fields.

III. CONCLUSION

In the above discussion we have developed a free-spacepropagation model for broadband optical fields based on aKarhunen-Lobve (KL) expansion of a coherence-separable,broadband field envelope. Owing to the linearity of thepropagation channel, each temporal mode could be propa-gated individually, and the broadband output field could becomputed as simply the superposition of all the individuallypropagated modes. Since the expansion coefficients of a KLexpansion are uncorrelated, the calculation of the spatialcorrelation of the output field was facilitated.

It was shown that for suitable broadband fields (spectralpurity -1) differential path length effects associated withbroadband field propagation could be combined into R. (r),

and written in terms of the first two derivatives of the tem-poral correlation RW (r). The broadband field correlationsEqs. (42) and (44) were found to reduce to well-known resultsfor quasimonochromatic source fields.

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7Note that this result is not precisely true because the exponentialsare not normalized.

8A. Papoulis, Probability, Random Variables, and Stochastic Pro-cesses (McGraw-Hill, New York, 1965).

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437 J. Opt. Soc. Am., Vol. 70, No. 4, April 1980 0030-3941/80/040437-06$00.50 Oc 1980 Optical Society of America 437