Department of Applied Mathematics, Research School of Physical Sciences, Australian National University,Canberra, ACT 2600, Australia
Received November 25, 1985; accepted February 19, 1986
A theory of optical coherence in terms of modal fields and the ensemble average of their amplitudes was recentlyplaced on a firm footing by Wolf [J. Opt. Soc. Am. 72,343 (1982)]. We discuss the physical consequences of this for-mulation for coherence definition and illustrate some special features of its use. The presence of an aperture andpartial reflection at a boundary demonstrate the effects on the degree of coherence of mode number reduction andselective modal attenuation. The propagation of fields from nonuniform sources is used to illustrate the advantageof using modes suited to propagation problems even though the modal correlations may be more involved than thoseintroduced by Wolf.
1. INTRODUCTION
The traditional theory of optical coherence1 is based oncorrelation functions for the optical fields. Recently,Wolf 24 has presented a new version of coherence theorythat is based on ensemble averages of modal components ofthe field. Although there are some difficult technical pointsin the derivation of the new theory, it is our contention thatits use allows the coherence properties of light fields to bedefined, understood, and studied with ease and a clearerunderstanding of the physical processes involved. We sup-port that contention with a brief discussion and some exam-ples.
Suppose that for frequency w the field may be describedby a set of terms am(w)4/m(r, w)exp(-iwt), where X/m are themodes and am(o) their amplitudes. 2A shows that co-herence theory can be developed in terms of the ensembleaverages (am(w)aj*(co)). Sources may be thought of interms of equivalent fields, e.g.,5 and initially Wolf speaksspecifically of modes for source fields but in fact also showsthat the theory holds for any field. This approach usingmodal amplitudes was exploited in work on diffraction,propagation problems, and optical fibers,610 but it was notuntil Wolf's work that the basic underlying theoretical con-structs were fully exposed.
In the theory developed by Wolf, the source field is shownto have a representation in terms of modes such that
(am(w)aj*(w)) = Cm5mj, (1)
where Cm is a constant. We shall assume fields and modesets for which this delta function result does not necessarilyhold. These modes may be chosen to be the natural modesfor describing propagation of a field, so that although theircorrelations may be more involved on occasions, the result-ing field at various points in space is more easily calculatedand understood. There is a trade-off to be made betweensimple correlation properties and suitability of the basismodes for carrying out calculations.
The coherence properties of the optical field are reducedto familiar terms' using the relations2 4
W(rj, r2 , w) = E (am(w)aj*(w)) .m(rl, w)¢Oj*(r2, C) (2)j,m
and
W(rj, r2 , C)
( 1W(rj, rl, co) W(r2 , r2 , W)](3)
where W is the cross-spectral density and g is the degree ofspatial coherence. The full field description involves anintegration over w to obtain the mutual coherence function rand intensity.2 4 In the rest of this paper we drop the explic-it dependence on with the implicit proviso that a finalfrequency integration or interpretation in terms of someoverall quasi-monochromatic scheme is required.
In Section 2 we make some general observations about theuse of Wolf's theory, and then in the following three sectionswe use it to explore the effect of apertures, transmissionthrough a surface, and propagation of fields from nonuni-form, partially coherent sources.
2. SOME GENERALITIES
The degree of coherence of an optical field is naturally ex-pressed in terms of the relative correlations between themodes used in its specification. Note that this actuallyintroduces two elements: the number of modes involvedand their interrelationship. For example, the field in anoptical fiber may be described in terms of a very few modesat reasonable distances from the source, and so a high degreeof coherence will be expected. We return to the question ofrestricting mode numbers in later sections with explicit ex-amples.
It seems appropriate to define the most incoherent field asthat having all modes equally and independently excited,i.e., Eq. (1) holds with Cm the same for all m. If this is torepresent total incoherence in some sense, it should be inde-pendent of the basis states used. Suppose that we have twocomplete sets of orthonormal basis functions, P/ml and bkm},and in the z = 0 plane, say, we write the total field as
The usual change of basis procedure, familiar in quantummechanics," leads to
(bjbj ,*) = E MjmMm,j,t(amam*). (4)m,m'
The matrix M is defined by
Mjm = JJ q5j*(,y)^m(xy)dxdy, (5)
and it has Hermitian conjugate Mt and MMt = unit matrix.Thus for total incoherence, substituting Eq. (1) with Cm = Cinto Eq. (4), we find the anticipated result:
(bjbj,*) = (ajaj,*) = Cbjj,. (6)
In practice the discrete sums may be supplemented by anintegration or continuous spectrum contribution," but theconclusion still holds. If not all the modes ipm are excited, orif Cm in Eq. (1) does depend on m, then Eq. (4) may lead tocorrelations among the 0j modes, i.e., (bjbjI*) may be non-zero when I # i'.
The above procedure is carried out in detail, including thevector or polarization effects, for a field exciting all modes onan optical fiber in Ref. 10, where it is shown that the appro-priate free-space incoherent field relates to a Lambertiansource. The normalization involved for modes actuallymeans unit power carried by the basis fields, and incoherentsources or fields are those in which the modes are indepen-dently excited and carry equal power.la This seems to besatisfactory from a physical point of view: all modes areequally weighted, and no information can be gained by mea-suring their relative powers when we have total incoherence.Notice that this definition can be used even for a limitedsubset of modes.
This definition of incoherence gives
W(rl, r 2) = Ej(r1)0j* (r2
and the degree to which this implies nonzero correlationsover short distances, as is the case in the usual conceptualview of an incoherent field, depends on the number of modesexcited and hence counted in the summation.
3. APERTURE EFFECTS
As an example of the role of mode-number reduction weexamine the effect of a finite source or an aperture. For thispurpose we again use the scalar field model, although thepolarization terms are easily included.la We consider wavespropagating into the half-space z > 0, and in the z = 0 planedefine the total field as
4(z = O) = JJ a(x) exp(iK R)dK.
As our example, we take the statistically homogeneous cwith all modes involved according to
(a(K)a*(K/)) = (K - K').
The vectors are
(9)
The wave vector is (kx, ky, k,) with magnitude k = /c = 2ir/Xfor wavelength X. For our case, kz = + (k2 - kX2-ky2)/2and any resulting complex values refer to evanescent ornonpropagating fields. The modes in question now are acontinuum of plane waves, and the degree of coherence, Eq.(3), for this field is
W(R1, R2, z1 = Z2 = 0) = (R1 - R2). (10)
This is of course an idealized mathematical example, asdiscussed in Ref. 12. All modes could never be involved inthis manner. When this field propagates away from the z =0 plane, the evanescent modes fall away and W changes, ofcourse.
We now introduce an aperture so that the field just leavingthe z = 0 plane is
(z = 0+) = 'f(z = 0-)S(R), (11)
where
S(R) = RI < a
= IRI > a. (12)
The effect of the aperture is demonstrated by calculatingW and ,u at a point near the z axis but a distance L from theaperture. We assume that a is large compared with wave-length; diffraction effects may be ignored, and so the aper-ture merely limits the angles of the waves involved, as shownin Fig. 1. (Diffraction effects may be important in someparts of the field; their inclusion would be straightforward,but messy, and would detract from the simple concept beingillustrated here.) The relevant values of the wave vector kare specified by IkJ < k cos Om, where Om is the angle shown inFig. 1. Using Eqs. (2) and (7)-(9), we find
W(R, R 2, z = L) = J||I<Km
where
t
exp[iK- (R1 -R2)]dK, (13)
I
aI-
(7)
ase
L Z
+Fig. 1. Geometry of aperture, radius a in z = 0 plane, and themaximum angle 0 m of waves relevant for observations near the axis
(8) at z = L.
Colin Pask
K = (k, k), R = x, y).
0 -_ -1h, -
Vol. 3, No. 7/July 1986/J. Opt. Soc. Am. A 1099
Km = k sin Am
=ka/L (L >> a). (14)
The integration is straightforward, and substituting into Eq.(3) gives the degree of coherence as
gU = 2J(KmR 12 )/(KmR 12 )
2J,(kaR1 2/L)/(kaR12/L), (15)
where R12 = IR, - R2 I and J1 is a Bessel function. Equation(15), usually derived in a different way, is called the vanCittert-Zernike theorem.'
The above calculation demonstrates explicitly how theaperture reduces the relevant mode number and its effect onthe coherence. The field at z = L becomes more coherent asa is reduced, or as L is increased, and fewer modes arerelevant. The part played by apertures in generating coher-ence was investigated by Wolf13 and Streifer,1 4 who intro-duced the use of resonator modes. The idea of a series ofapertures13"14 leads to consideration of a continuous apertureor bounded region,15"16 and the enhancement of coherencewith propagation was demonstrated experimentally.'5"16
This problem has been reexamined recently using Wolf'snew coherence theory.' 7 The development of coherence inwaveguides such as optical fibers has now been extensivelyinvestigated theoretically' 8 2 ' and experimentally.2 2 -2 4
4. TRANSMISSION THROUGH AN INTERFACE
The discussion in Section 2 concerned the change from onemode set to another, and we now give the details for such aproblem, which occurs at the interface between two half-spaces. If the mode change in Section 2 referred to a prob-lem of this type, it assumed that the field passed through theinterface undistorted, which is an idealization. Here weallow for a reflected component, so that the transmittedmodes are now selectively attenuated, and hence a change incoherence is expected.
We assume uniform media with refractive index nj for z <0 and n2 for z > 0, with n2 > n1, and the total field incidentfrom z < 0 is
inc = f< a(K)expli[K R + (k,2 -K2
)12
z]I dK, (16)
where K and R are defined by Eqs. (9). Since ki = 27rni/X,where X is the wavelength in vacuum, we are considering allpropagating waves. We also assume the statistical propertydefined by Eq. (8), which means that for z < 0 the incidentfield has in any z = constant plane the degree of coherence
Ainc = 2Jj(kjR1 2 )/(kR1 2 ). (17)
This of course would be the coherence for z > 0 if there wereno interface, i.e., n2 = nj.
Each plane wave is transmitted through the interface'according to Snell's law (K is unchanged) and Fresnel's law[the transmitted wave has amplitude reduced by a factorT(K)]. The exact transmission coefficient depends on polar-ization, but as a reasonable model to demonstrate physicalconsequences we take
T2(K) = T02 cos 01 = T_2 [(1 -K2/kl2)]/2,
with the z axis and To is less than 1. Thus the total trans-mitted field is
=trans f J T(K)a(K)expli[K R + (k22 - K2)/ 2 z]I dK,
(19a)
where k2 = 2n 2/X. Applying Eqs. (2) and (3) leads to thedegree of coherence in the transmitted field in any z =constant plane:
Atrans = 3J3/2(kjR12)AkjR12)3/2, (19b)
where J3 /2 is the fractional-order Bessel function.25 Thedegree of coherence does not depend on To, but the relativechange of modal amplitudes has caused a modification. Ifwe define the coherence length Rcoh as that value of R12 forwhich A first becomes zero, we find
Rcohinc = 3.83/ki,
Rcoh,trans = 4.49/kl.
(20a)
(20b)
If we used a different transmission coefficient, rather thanEq. (18), then the coherence length would change, but wewould still expect to find Rcohtrans > Rcohinc.
Note that if T(K) = T so that all waves are similarlyreflected, the coherence properties of the transmitted fieldare given by Eqs. (17) and (20a) since the modal volume hasnot been reduced or selectively weighted.
5. A PROPAGATION EXAMPLE
In Section 2 we referred to the problem of the field launchedinto an optical fiber by a particular source. In this sectionwe consider as an example the same problem but with thefiber replaced by free space. Thus the appropriate modesfor solving the propagation problem are plane waves, and weshall therefore describe the initial conditions or source interms of those modes. This contrasts with the generalscheme of Wolf,2 4 in which modes used to describe thesource are derived from an integral equation dictated by thesource specification. Starikov and Wolf 26 carry out thatprocedure for Gaussian Schell-model sources, and we alsouse that familiar and interesting example.27-29 Althoughthe final results are well known, this example clearly showsthe power and simplicity of a new description based entirelyon the propagation modes.
Mt(R1, R 2, z = 0) = exp[-(po 2- O 2 )R2 2/4}. (29)
Thus we have shown that the correlation for plane-wavemodes expressed by Eqs. (22), (25), and (26) leads to aGaussian Schell-model source; the modes for this source,which have correlation as in Eq. (1), are found by Starikovand Wolf.2 6
The advantage of the present formulation is that the fieldeverywhere is now immediately given by Eq. (21). The formof Eq. (24) and the results in Ref. 30 also mean that the far-field radiant intensity J(0) is obtained directly as
J(0) = go(27rk cos 0)2 exp(-k 2 sin 2 0/po 2 )' (30)
where 0 is the angle between the source-to-far-field observa-tion point and the z axis. We observe that sources of thetype under consideration all have the same far-zone intensi-ty distributions if they have the same values for k/po andgok2; these are exactly the same results as in Theorem 1 ofRef. 27 apart from a notational change.
The far-field J(0) is simply related to the source in theabove specification using plane-wave modes. In general,
J(0) = (27rk cos 0)2 f(k s,)g(O), (31)
where sI is the projection onto the source plane of the unitvector in the direction from the source to far-field observa-tion point.30 We immediately see that the components off(p) for IPI > k do not effect the far field and that the detailsof the functiong are not revealed. The relationship betweensource and far field has been investigated intensely; see Ref.26 for review. The function f(p) specifies the mode volumeinvolved, and a reduced volume leads to a more directed farfield, as evidenced by Eq. (31).
6. CONCLUSION
Although the source coherence and intensity propertiesspecify a set of modes with simple amplitude relations,2 4 itmay be useful to employ modes directly related to the propa-gation problem. In the former case the modes will vary asthe source does, whereas in the latter case the modes remain
the same but the amplitude relations vary. The above ex-amples demonstrate the use of the propagating-mode pic-ture that readily allows one to observe the effects of modevolume and attenuation on optical coherence.
7. ACKNOWLEDGMENT
The author thanks the reviewers for comments used to cor-rect and improve this paper.
* Present address, Department of Mathematics, Universi-ty College, Australian Defence Force Academy, Campbell,ACT 2600, Australia.
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