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A type of source with invariant far zone spectrum
Volume 66, number 1
OPTICS COMMUNICATIONS
I Aprtl 1988
A TYPE OF SOURCE WITH INVARIANT FAR ZONE SPECTRUM
F. G O R I , G. G U A T T A R I , C. P A L M A Dzparttmento dt Fzstca, Umversttd dt Roma "La Sapwnza'" P le Aldo Moro, 2, 00185 Roma, Italy and Gruppo Naztonale dt Struttura della Materta, CNR, Italy
and
C. P A D O V A N I Dtparttmento dl Ingegnerta Elettrwa, Untverstt,~ dt L "Aquda, Monteluco dz Roto, L "Aquda, Italy
Received 9 October 1987
A type of parhally coherent source derived from the well known Collett-Wolf Schell model source is considered. It Is shown that there are many ways to choose the source parameters so as to ensure that the normahzed field spectrum m the far zone is independent of the direction. Depending on the choice of those parameters we can pass from completely spatmUycoherent sources to nearly incoherent sources while preserving the spectrum lnvanance. In a certain hmit, the source becomes quas~-homogeneous and thus obeys the Wolf's scahng law [ E. Wolf, Phys. Rev. Lett. 56 (1986) 1370]
1. Introduction
In spite o f the widely held belief that the spectrum in a light field is i n d e p e n d e n t o f the observation point, in general the s p e c t r u m varies from one p o i n t to another. In particular, in the far zone o f a source the spectrum m a y well d e p e n d on the direction o f observation. In a recent p a p e r [ l ], W o l f showed that for quasi-homogeneous sources the n o r m a l i z e d spectrum is invariant throughout the far zone i f the degree o f spectral coherence satisfies a certain scaling law. Violations o f the scaling law lead to far zone spectra d e p e n d i n g on the direction o f observation [ 2 ]. As underlined by W o l f in his paper, the scaling law applies to q u a s i - h o m o g e n e o u s sources a n d further investigations are required for other sources. F o r example, it has been shown [ 3 ] that for a c o m m o n l y used source, n a m e l y for a laser, the spectrum varies across any cross-section o f the light beam. We then ask whether there is any class o f sources not enc o m p a s s e d by the q u a s i - h o m o g e n e o u s m o d e l for which c o n d i t i o n s ensuring spectrum invariance in the far zone can be found. In this paper, we consider a type o f source d e r i v e d from the well-known Collett-Wolf Schell m o d e l source extensively s t u d i e d in the literature [ 4 - 2 2 ]. We will show that, u n d e r suitable conditions, such a source radiates a field whose n o r m a l i z e d s p e c t r u m is i n v a r i a n t throughout the far zone. The a b o v e conditions can be met in several ways so that the resulting source can have vastly different properties o f spatial coherence. We will also show that there exists a certain limiting case in which the source becomes quasi-homogeneous [4] a n d the conditions for spectrum i n v a r i a n c e coincide with the W o l f ' s scaling law. 0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume 66, number 1
OPTICS COMMUNICATIONS
1 April 1988
2. Description of the source and statement of the problem Let us consider a partially coherent planar source. We assume that for any two points with position vectors r~ and r2 in the source plane the cross spectral density [ 23 ], at temporal frequency v, say Wo (rl, r2, v), is given by W o ( r l , r2, v) =Ao(v) exp[iy(v) (r 2 - r 22) __p2(/j) ( r 1 + r 2 )2 _ m2(/)) ( r , - - r 2 ) 2 ] .
(2.1)
Here, y(v) is a real function of v. The functions Ao(v), p(v) and m(v) are all real positive. We shall also use the notations
7(v) =nv/cR(v),
p2(v) = 1/[8a~(v)] , m2(v) = l/[8a2(v)] + 1/[2a2(v)] ,
(2.2)
where R (v) is a real function and where a i (v) and a , ( v ) are real positive functions. The meaning of all these functions will be shortly discussed. As implied by eq. (2.2), we assume that the following inequality holds, (2.3)
m2(p) >p2(v) .
The power spectrum [ 23 ], which is also called the optical intensity at frequency v, at any point r of the source is obtained from eq. (2.1) letting r~ = r a = r . This gives
Wo(r,r, v) =Ao(v) e x p [ - 4 p E ( v ) r E] =Ao(v) exp{-r2/[2a~(v)]}.
(2.4)
It is seen that a~(v) gives a measure of the effective radius of the source at frequency v. The complex degree of spectral coherence [23], defined as follows
lto(rl, r2, v) = Wo(rl, r2, v)/[ Wo(rl, rl, v) Wo(r2, r2, v)] 1/2 ,
(2.5)
turns out to be go(r1, r2, v) = exp{iT(v) (r~ -r22) - [m2(v) -p2(v)](rx --rE) 2} = exp{inv(r~ -r2)/[cR(v)] - (rl - r 2 ) 2 / [ 2 a 2 ( v ) ] }.
(2.6)
It is seen from above that, an any frequency v, the source is quite similar to a Collett-Wolf Schell model source [ 6 ]. The only difference is the presence of the quadratic phase factor. As a matter of fact, the cross-spectral density (2.1) is of the form pertaining to the field radiated by a Collett-Wolf Schell model source after the field has propagated over a certain distance [ 17 ]. At any frequency v, R (v) is the wavefront radius of curvature of the underlying modes of the source [17,24,25], whereas tru(v) gives a measure of the correlation length across the source. It is to be noted that, because of the phase factor, the functional form of eq. (2.1) is no longer of the type required by the Schell model [26]. Let us adtl a further comment. When considered as a function of r, the power spectrum is in general a gaussian curve (eq. (2.4)). As far as the dependence on v is concerned, the power spectrum is a different function of v at each source point. However, i f p ( v ) (and then th ( v )) is independent of v, the power spectrum is one and the same function of v across the whole source except for an amplitude factor. We now state the spectrum invariance problem. Let us consider the field spectrum in the far zone. Following Wolf [ 1 ], we shall refer to the normalized spectrum, say S (°°) (u, v), where u is the unit vector specifying the direction of observation. This function is defined as follows
S '~°, ( a , v ) = J ( u , v ) / f J ( u , v ) d r ,
(2.7)
o
where J(u, v) is the radiant intensity at frequency v [27]. For sources of any state of coherence, the radiant intensity can be evaluated in the following way [ 27 ]
Volume 66, number 1
v~ cos20 ~f c----T---
J(u,v)=
OPTICS COMMUNICATIONS
Wo(rl,r2,v) exp[--27~i(v/c)u± " ( r t - - r 2 ) ] d2r! d2r2 ,
1 April 1988
(2.8)
the integrals being extended to the source plane. Here, 0 is the angle between the direction of u and the normal to the source plane, u± is the vector obtained through orthogonal projection of u on the source plane and c is the speed of light in the (supposedly homogeneous) medium in which the field propagation occurs. It is clear from eqs. (2.7) and (2.8) that the normalized spectrum will be in general a different function of v for each direction of observation, i.e. for each choice of the unit vector u. We briefly say that the spectrum is invariant in the far zone if the function S(~)(u, v) is actually independent of u. In ref. [ 1 ], Wolf proved that for a quasi-homogeneous source the spectrum is invariant if the following scaling law is obeyed by the degree of spectral coherence across the source
m( rt , r2, v) = f [v( r2 - r , )] ,
(2.9)
i.e. if the degree of spectral coherence is a function of the variable v(r2-rl) only. We now ask whether the functions y(v), p(v) and m(v) can be chosen in such a way as to ensure that the normalized spectrum is invariant throughout the far zone.
3. The condition for spectrum invariance Let us evaluate the radiant intensity (2.8) when the cross spectral density of the source is given by eq. (2.1). We shall introduce the variables
s=(rl+r2)/2,
t=rl-r2.
(3.1)
In terms of the new variables, the radiant intensity produced by our source is given by
j(u,v)=Ao(v) v2 c2 c°s2Of ~ exp[2i~,(v) s.t-4p2(p) s 2 --m2(v) t 2 --2ni(v/C) U±"t] dZsd2t.
(3.2)
For the sake of brevity, we omit the explicit evaluation of the integral of eq. (3.2). The result is as follows
J(u,v) =
A°(v) nEv2 c°s20 ( 4~2p2(v) v2 ) c214pZ(v ) mE(v)+~2(v)] exp - c214p2(v ) me(v)+yz(v) ] u~ .
(3.3)
We now require that the normalized spectrum (2.7) is independent of u. It is seen from eqs. (2.7) and (3.3) that this requirement is satisfied if the gaussian function appearing in eq. (3.3) does not depend on v. Accordingly, the following condition for spectrum invariance is found
47r2p2(v) V2/C214p2(p)m2(v)+~2(V)]
=
1/2d 2 ,
(3.4)
where the dimensionless positive constant d can be taken as a measure of the width for the gaussian function appearing in eq. (3.3). When eq. (3.4) holds, the normalized spectrum becomes oo
=
tAo( )/p2( )J/j t -
(3.5)
O
everywhere in the far zone. On substituting from eq. (2.2) into eq. (3.4) and rearranging terms, the spectrum invariance condition can be written
Volume 66, number 1 1
1 April 1988
4n2a2(v) v2 = 4ha6 2 v2 .
1
4a~(v----~ + ~
OPTICS COMMUNICATIONS
+ c2n2(v)
c2
(3.6)
We note that this condition is to be satisfied for those values of v for which Ao(v) is different from zero. In any practical case, this means that a finite range of values of v is involved. In order to satisfy eq. (3.6), we can rely on the choice of three different functions, namely R ( v ) , at(v) and a~,(v). This gives us a considerable freedom in making our choices. We shall limit ourselves to a few examples, assuming from now on that a,(v) is given a constant value, say 2Z, at any frequency, i.e. ai(v) =27=const.
(3.7)
We already noted that in this case the power spectrum is the same (up to a multiplicative term) at any source point, being everywhere proportional to Ao(v). We now see from eq. (3.5) that the normalized spectrum in the far zone is also proportional to Ao(v). In other words, measurements of the spectrum in the far zone give the same results that would be obtained at the source. Although a~(v) has now been fixed, a~(v) and R ( u ) are still at our disposal in order to satisfy eq. (3.6). For this reason, completely different properties of spatial coherence are compatible with the spectrum invariance condition. For example, a complete coherence limit exists in which a ~ ~ . In this case, eq. (3.6) requires that R ( v ) is the following function of
2nXv R ( v ) = +_ C[(47[2¢~2/C2) V2 -- 1/4272 ] 1/2
"
(3.8)
Of course, the value assigned to 27 must ensure that the expression under the square root symbol is non negative for the values of v for which Ao (v) # 0. A comment about the double sign in eq. (3.8) is in order. As we already said, the presence of a phase factor in the cross spectral density (2.1) can be traced back to the curvature of the wavefronts of the underlying natural modes. Thus, a change of sign of R ( v ) corresponds to reversing the curvature of those wavefronts (from diverging to converging). Both signs are equally valid. This can be understood by remembering that our observations are made in the far zone so that it does not matter whether the wavefronts of the modes diverge from the source onwards, or they first converge to some waist (at a finite distance from the source) and then diverge. The present example shows, in a particular case, that the modulus of the degree of spectral coherence can be the same at all frequencies. Inspection of eq. (3.6) reveals that this can be true even if the hypothesis a~-~ ao is removed. In other words, we can devise a source whose modulus of the degree of spectral coherence has the same finite width at any frequency and whose spectrum satisfies the invariance condition. As another significant example, let us choose R (u)-~ oo. In this case, eq. (2.1) describes a Collett-Wolf Schell model source. Eq. (3.7) now gives a , ( v ) = [(4~t2~2/c2) v 2 - 1/4272] -1:2
(3.9)
It is seen that ea must vary with v if eq. (3.9) has to be satisfied. This of course is due to the fact that both al and R have been given a constant value. The source becomes quasi-homogeneous if, in addition to R-.oo, the hypothesis cr,<<27 is made. In this case, it is easily seen that eq. (3.9) is replaced by
a~( v ) =c/2n~v .
(3.10)
The degree of spectral coherence (2.6) becomes now /zo(ri, rE, V) = e x p [ -- 2n2~2v2(rl - - r 2 )2/c2] , in complete agreement with the Wolf's scaling law (2.9).
(3.1 1 )
Volume 66, number 1
OPTICS COMMUNICATIONS
1 April 1988
4. Conclusions The Collett-Wolf Schell model source has proved to be of interest from several points of view. In this paper, we have seen that a generalized version of it has remarkable properties in connection to the phenomenon of spectrum propagation. The analysis o f these properties may give hints about propa.gation phenomena from other partially coherent sources. In addition, the now well established knowledge of the modes o f the Collett-Wolf Schell model source may be of help in establishing a link between mode propagation and power spectrum propagation.
References [ 1 ] E. Wolf, Phys. Rev. Lett. 56 (1986) 1370. [2] G.M. Morris and D. Faklis, Optics Comm. 62 (1987) 5. [3] F. Gon and R. Orella, Optlcs Comm. 49 (1984) 173. [4] W.H. Carter and E. Wolf, J. Opt. Soc. Am. 67 (1977) 785. [ 5 ] E. Collett and E. Wolf, Optics Lett. 2 (1978) 27 [6] E. Wolf and E. Collett, Optics Comm. 25 (1978) 293. [7] J.T. Foley and M.S. Zubairy, Optics Comm. 26 (1978) 297. [8] P. De Santls, F. Gon, G. Guattan and C. Palma, Optics Comm. 29 (1979) 256. [9] B.E A. Saleh, Optics Comm. 30 (1979) 135. [ 10] J D Fanna, L.M. Narduccl and E. Collett, OpUcs Comm. 32 (1980) 203. [ 11 ] F. Got1, OpUes Comm. 34 (1980) 301. [ 12] A.T. Friber8 and R.J. Sudol, OpUcs Comm. 41 (1982) 383. [ 13] Y. L~ and E. Wolf, Optics Lett. 7 (1982) 256. [14] A. Starikov and E. Wolf, J. Opt. Soc. Am. 72 (1982) 923. [ 15] A. Starikov, J. Opt. Soc. Am. 72 (1982) 1538. [ 16 ] J Deschamps, D. Courjon and J. Bulabols, J. Opt. Soc. Am. 73 (1983) 256. [ 17] F. Gon, Optics Comm. 46 (1983) 149. [ 18] F. Gon and G. Guattan, Optics Comm. 48 (1983) 7. [ 19 ] A.T. Friberg and R.J. Sudol, OpUca Acta 30 (1983 ) 1075. [20] R Simon, E C.G. Sudarshan and N. Mukunda, Phys. Rev. A 29 (1984) 3273. [21 ] P De Santls, F. Gon, G. Ouattan and C. Palma, Optlca Acta 33 (1986) 315. [22] M.S. Zubairy and J.K. McIver, Phys. Rev. A 36 (1987) 202. [23] L. Mandel and E. Wolf, J. Opt. Soc. Am. 66 (1986) 529. [24] E. Wolf, J. Opt. Soc. Am. 72 (1982) 343. [25] E. Wolf, J. Opt. Soc Am. A 3 (1986) 76. [26] A C. Schell, IEEE Trans. Antennas Propag. 15 (1967) 187. [27] E.W. Marchand and E Wolf, J. Opt. Soc. Am. 64 (1974) 1219.
OPTICS COMMUNICATIONS
I Aprtl 1988
A TYPE OF SOURCE WITH INVARIANT FAR ZONE SPECTRUM
F. G O R I , G. G U A T T A R I , C. P A L M A Dzparttmento dt Fzstca, Umversttd dt Roma "La Sapwnza'" P le Aldo Moro, 2, 00185 Roma, Italy and Gruppo Naztonale dt Struttura della Materta, CNR, Italy
and
C. P A D O V A N I Dtparttmento dl Ingegnerta Elettrwa, Untverstt,~ dt L "Aquda, Monteluco dz Roto, L "Aquda, Italy
Received 9 October 1987
A type of parhally coherent source derived from the well known Collett-Wolf Schell model source is considered. It Is shown that there are many ways to choose the source parameters so as to ensure that the normahzed field spectrum m the far zone is independent of the direction. Depending on the choice of those parameters we can pass from completely spatmUycoherent sources to nearly incoherent sources while preserving the spectrum lnvanance. In a certain hmit, the source becomes quas~-homogeneous and thus obeys the Wolf's scahng law [ E. Wolf, Phys. Rev. Lett. 56 (1986) 1370]
1. Introduction
In spite o f the widely held belief that the spectrum in a light field is i n d e p e n d e n t o f the observation point, in general the s p e c t r u m varies from one p o i n t to another. In particular, in the far zone o f a source the spectrum m a y well d e p e n d on the direction o f observation. In a recent p a p e r [ l ], W o l f showed that for quasi-homogeneous sources the n o r m a l i z e d spectrum is invariant throughout the far zone i f the degree o f spectral coherence satisfies a certain scaling law. Violations o f the scaling law lead to far zone spectra d e p e n d i n g on the direction o f observation [ 2 ]. As underlined by W o l f in his paper, the scaling law applies to q u a s i - h o m o g e n e o u s sources a n d further investigations are required for other sources. F o r example, it has been shown [ 3 ] that for a c o m m o n l y used source, n a m e l y for a laser, the spectrum varies across any cross-section o f the light beam. We then ask whether there is any class o f sources not enc o m p a s s e d by the q u a s i - h o m o g e n e o u s m o d e l for which c o n d i t i o n s ensuring spectrum invariance in the far zone can be found. In this paper, we consider a type o f source d e r i v e d from the well-known Collett-Wolf Schell m o d e l source extensively s t u d i e d in the literature [ 4 - 2 2 ]. We will show that, u n d e r suitable conditions, such a source radiates a field whose n o r m a l i z e d s p e c t r u m is i n v a r i a n t throughout the far zone. The a b o v e conditions can be met in several ways so that the resulting source can have vastly different properties o f spatial coherence. We will also show that there exists a certain limiting case in which the source becomes quasi-homogeneous [4] a n d the conditions for spectrum i n v a r i a n c e coincide with the W o l f ' s scaling law. 0 030-4018/88/$03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
Volume 66, number 1
OPTICS COMMUNICATIONS
1 April 1988
2. Description of the source and statement of the problem Let us consider a partially coherent planar source. We assume that for any two points with position vectors r~ and r2 in the source plane the cross spectral density [ 23 ], at temporal frequency v, say Wo (rl, r2, v), is given by W o ( r l , r2, v) =Ao(v) exp[iy(v) (r 2 - r 22) __p2(/j) ( r 1 + r 2 )2 _ m2(/)) ( r , - - r 2 ) 2 ] .
(2.1)
Here, y(v) is a real function of v. The functions Ao(v), p(v) and m(v) are all real positive. We shall also use the notations
7(v) =nv/cR(v),
p2(v) = 1/[8a~(v)] , m2(v) = l/[8a2(v)] + 1/[2a2(v)] ,
(2.2)
where R (v) is a real function and where a i (v) and a , ( v ) are real positive functions. The meaning of all these functions will be shortly discussed. As implied by eq. (2.2), we assume that the following inequality holds, (2.3)
m2(p) >p2(v) .
The power spectrum [ 23 ], which is also called the optical intensity at frequency v, at any point r of the source is obtained from eq. (2.1) letting r~ = r a = r . This gives
Wo(r,r, v) =Ao(v) e x p [ - 4 p E ( v ) r E] =Ao(v) exp{-r2/[2a~(v)]}.
(2.4)
It is seen that a~(v) gives a measure of the effective radius of the source at frequency v. The complex degree of spectral coherence [23], defined as follows
lto(rl, r2, v) = Wo(rl, r2, v)/[ Wo(rl, rl, v) Wo(r2, r2, v)] 1/2 ,
(2.5)
turns out to be go(r1, r2, v) = exp{iT(v) (r~ -r22) - [m2(v) -p2(v)](rx --rE) 2} = exp{inv(r~ -r2)/[cR(v)] - (rl - r 2 ) 2 / [ 2 a 2 ( v ) ] }.
(2.6)
It is seen from above that, an any frequency v, the source is quite similar to a Collett-Wolf Schell model source [ 6 ]. The only difference is the presence of the quadratic phase factor. As a matter of fact, the cross-spectral density (2.1) is of the form pertaining to the field radiated by a Collett-Wolf Schell model source after the field has propagated over a certain distance [ 17 ]. At any frequency v, R (v) is the wavefront radius of curvature of the underlying modes of the source [17,24,25], whereas tru(v) gives a measure of the correlation length across the source. It is to be noted that, because of the phase factor, the functional form of eq. (2.1) is no longer of the type required by the Schell model [26]. Let us adtl a further comment. When considered as a function of r, the power spectrum is in general a gaussian curve (eq. (2.4)). As far as the dependence on v is concerned, the power spectrum is a different function of v at each source point. However, i f p ( v ) (and then th ( v )) is independent of v, the power spectrum is one and the same function of v across the whole source except for an amplitude factor. We now state the spectrum invariance problem. Let us consider the field spectrum in the far zone. Following Wolf [ 1 ], we shall refer to the normalized spectrum, say S (°°) (u, v), where u is the unit vector specifying the direction of observation. This function is defined as follows
S '~°, ( a , v ) = J ( u , v ) / f J ( u , v ) d r ,
(2.7)
o
where J(u, v) is the radiant intensity at frequency v [27]. For sources of any state of coherence, the radiant intensity can be evaluated in the following way [ 27 ]
Volume 66, number 1
v~ cos20 ~f c----T---
J(u,v)=
OPTICS COMMUNICATIONS
Wo(rl,r2,v) exp[--27~i(v/c)u± " ( r t - - r 2 ) ] d2r! d2r2 ,
1 April 1988
(2.8)
the integrals being extended to the source plane. Here, 0 is the angle between the direction of u and the normal to the source plane, u± is the vector obtained through orthogonal projection of u on the source plane and c is the speed of light in the (supposedly homogeneous) medium in which the field propagation occurs. It is clear from eqs. (2.7) and (2.8) that the normalized spectrum will be in general a different function of v for each direction of observation, i.e. for each choice of the unit vector u. We briefly say that the spectrum is invariant in the far zone if the function S(~)(u, v) is actually independent of u. In ref. [ 1 ], Wolf proved that for a quasi-homogeneous source the spectrum is invariant if the following scaling law is obeyed by the degree of spectral coherence across the source
m( rt , r2, v) = f [v( r2 - r , )] ,
(2.9)
i.e. if the degree of spectral coherence is a function of the variable v(r2-rl) only. We now ask whether the functions y(v), p(v) and m(v) can be chosen in such a way as to ensure that the normalized spectrum is invariant throughout the far zone.
3. The condition for spectrum invariance Let us evaluate the radiant intensity (2.8) when the cross spectral density of the source is given by eq. (2.1). We shall introduce the variables
s=(rl+r2)/2,
t=rl-r2.
(3.1)
In terms of the new variables, the radiant intensity produced by our source is given by
j(u,v)=Ao(v) v2 c2 c°s2Of ~ exp[2i~,(v) s.t-4p2(p) s 2 --m2(v) t 2 --2ni(v/C) U±"t] dZsd2t.
(3.2)
For the sake of brevity, we omit the explicit evaluation of the integral of eq. (3.2). The result is as follows
J(u,v) =
A°(v) nEv2 c°s20 ( 4~2p2(v) v2 ) c214pZ(v ) mE(v)+~2(v)] exp - c214p2(v ) me(v)+yz(v) ] u~ .
(3.3)
We now require that the normalized spectrum (2.7) is independent of u. It is seen from eqs. (2.7) and (3.3) that this requirement is satisfied if the gaussian function appearing in eq. (3.3) does not depend on v. Accordingly, the following condition for spectrum invariance is found
47r2p2(v) V2/C214p2(p)m2(v)+~2(V)]
=
1/2d 2 ,
(3.4)
where the dimensionless positive constant d can be taken as a measure of the width for the gaussian function appearing in eq. (3.3). When eq. (3.4) holds, the normalized spectrum becomes oo
=
tAo( )/p2( )J/j t -
(3.5)
O
everywhere in the far zone. On substituting from eq. (2.2) into eq. (3.4) and rearranging terms, the spectrum invariance condition can be written
Volume 66, number 1 1
1 April 1988
4n2a2(v) v2 = 4ha6 2 v2 .
1
4a~(v----~ + ~
OPTICS COMMUNICATIONS
+ c2n2(v)
c2
(3.6)
We note that this condition is to be satisfied for those values of v for which Ao(v) is different from zero. In any practical case, this means that a finite range of values of v is involved. In order to satisfy eq. (3.6), we can rely on the choice of three different functions, namely R ( v ) , at(v) and a~,(v). This gives us a considerable freedom in making our choices. We shall limit ourselves to a few examples, assuming from now on that a,(v) is given a constant value, say 2Z, at any frequency, i.e. ai(v) =27=const.
(3.7)
We already noted that in this case the power spectrum is the same (up to a multiplicative term) at any source point, being everywhere proportional to Ao(v). We now see from eq. (3.5) that the normalized spectrum in the far zone is also proportional to Ao(v). In other words, measurements of the spectrum in the far zone give the same results that would be obtained at the source. Although a~(v) has now been fixed, a~(v) and R ( u ) are still at our disposal in order to satisfy eq. (3.6). For this reason, completely different properties of spatial coherence are compatible with the spectrum invariance condition. For example, a complete coherence limit exists in which a ~ ~ . In this case, eq. (3.6) requires that R ( v ) is the following function of
2nXv R ( v ) = +_ C[(47[2¢~2/C2) V2 -- 1/4272 ] 1/2
"
(3.8)
Of course, the value assigned to 27 must ensure that the expression under the square root symbol is non negative for the values of v for which Ao (v) # 0. A comment about the double sign in eq. (3.8) is in order. As we already said, the presence of a phase factor in the cross spectral density (2.1) can be traced back to the curvature of the wavefronts of the underlying natural modes. Thus, a change of sign of R ( v ) corresponds to reversing the curvature of those wavefronts (from diverging to converging). Both signs are equally valid. This can be understood by remembering that our observations are made in the far zone so that it does not matter whether the wavefronts of the modes diverge from the source onwards, or they first converge to some waist (at a finite distance from the source) and then diverge. The present example shows, in a particular case, that the modulus of the degree of spectral coherence can be the same at all frequencies. Inspection of eq. (3.6) reveals that this can be true even if the hypothesis a~-~ ao is removed. In other words, we can devise a source whose modulus of the degree of spectral coherence has the same finite width at any frequency and whose spectrum satisfies the invariance condition. As another significant example, let us choose R (u)-~ oo. In this case, eq. (2.1) describes a Collett-Wolf Schell model source. Eq. (3.7) now gives a , ( v ) = [(4~t2~2/c2) v 2 - 1/4272] -1:2
(3.9)
It is seen that ea must vary with v if eq. (3.9) has to be satisfied. This of course is due to the fact that both al and R have been given a constant value. The source becomes quasi-homogeneous if, in addition to R-.oo, the hypothesis cr,<<27 is made. In this case, it is easily seen that eq. (3.9) is replaced by
a~( v ) =c/2n~v .
(3.10)
The degree of spectral coherence (2.6) becomes now /zo(ri, rE, V) = e x p [ -- 2n2~2v2(rl - - r 2 )2/c2] , in complete agreement with the Wolf's scaling law (2.9).
(3.1 1 )
Volume 66, number 1
OPTICS COMMUNICATIONS
1 April 1988
4. Conclusions The Collett-Wolf Schell model source has proved to be of interest from several points of view. In this paper, we have seen that a generalized version of it has remarkable properties in connection to the phenomenon of spectrum propagation. The analysis o f these properties may give hints about propa.gation phenomena from other partially coherent sources. In addition, the now well established knowledge of the modes o f the Collett-Wolf Schell model source may be of help in establishing a link between mode propagation and power spectrum propagation.
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