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Some relationships between the correlation coefficients of planar sources and of their far fields
Volume 25, number 3
OPTICS COMMUNICATIONS
June 1978
SOME RELATIONSHIPS BETWEEN THE CORRELATION COEFFICIENTS OF PLANAR SOURCES AND OF THEIR FAR FIELDS William H. CARTER
Naval Research Laboratory, Washington, D.C. 20375, USA and Emil WOLF
Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, N.Y. 14627, USA Received 8 February 1978 Two new correlation coefficients are introduced, that characterize the correlations between tbe spatial frequency components of the ,source distribution and of the field distribution respectively, in the plane of a two-dimensional source. It is then shown that except possibly for a simple phase factor depending on the magnitudes of the .spatial frequencies, the two correlation coefficients are equal to each other. It is further shown that the degree of spatial coherence of the far field generated by the ,source is equal, except possibly for a simple geometrical phase factor, to either of the two new correlation coefficients relating to the distributions in the source plane.
1. Introduction In a recent paper [1] a certain relationship was established between the cross-spectral density functions of the source distribution and of the field distribution in the plane of a primary two-dimensional source. In the present paper we show, with the help of this result, that there exists a simple relationship between certain new correlation coefficients that characterize the correlations between the spatial frequency components of the source distribution and of the field distribution respectively in the plane of the source. In fact, the two correlation coefficients are found to be equal, except possibly for a simple phase factor depending on the magnitudes of the spatial frequencies. We also show that the degree of spatial coherence of the far field generated by the source is equal, except possibly for a simple phase factor depending on geometry, to either of the two new correlation coefficients relating to the distributions in the source plane.
2. Field correlation functions and source correlation functions We consider an optical field V(r, t), generated by a primary source distribution Q(r, t) in free space. Both V(r, t) and Q(r, t) are taken to be scalar functions of position (r) and time (t). We assume that the source distribution Q(r, t) is localized, for all time, in some finite domain around the origin r = 0. We take V(r, t) and Q(r, t) to be analytic signals :[: and we consider them to be stationary random variables. They are coupled by the inhomogeneous wave equation V2 V(r, t) - 1
~2 V(r, t) . . . . 4~Q(r, t), c 2 ~t 2
(2.1)
where c is the vacuum speed of light. Let
1
v(r, ~ ) = ~ f
V(r,t)e it°t dt,
(2.2)
$ For the definition of analytic signals see, for example, Born and Wolf {21. 288
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and
,y Q(r,t)e i°~t dt
p(r, co) = ~
~(0)¢~ l ffw~O)(rl,r2,60) o 'J1'1'2' 60) = (2n)4 (2.3)
--oo
be the Fourier transforms of V(r,t) and of Q(r,t) respectively. As is well known from the theory of stationary random processes the Fourier transforms (2.2) and (2.3) do not exist as ordinary integrals and must be interpreted in the sense of the theory of generalized functions. Let us now introduce the cross-spectral density functions Wo(r1,r2, 6o) and Wa(rl, r2, co) of the field- and the source variable respectively, via the relations (e.g. 13] )
(v(rl, 60)v*(r2,60')) = Wv(rl, r2,60)5(w - co'),
June 1978
(2.4)
(p(r a,co)p*(r2,w' )) = Wo(r 1 , r 2 , w ) 5 ( c o - co'). (2.5) The Dirac delta function 5 appears on the right-handsides of these equations as consequence of the assumed stationarity of the field and of the source. It is clear from the above formulae that Wv(rl, r2, w) represents the correlation that exists between the (generalized) Fourier components of frequency 60 of the light oscillations at points r 1 and r 2 in the field. Wo(rl, r2, 60) has, of course, a similar significance with respect to the source oscillations. Suppose now that the source is two-dimensional, occupying a finite portion of the plane z = 0 and that it radiates into the half-space z > 0. We will indicate by a superscript zero values of quantities pertaining to the source plane. Thus, for example u(0)(rl, w) indicates the values of o(rl, 60) when the point specified by the position vector r 1 is located in the source plane z = O. W(°)(rl, rE, 60) indicates the value of Wo(rl, r2, co) when both the points, specified by the position vectorsr 1 and r2, are located in this plane, etc. Now as a consequence of the fact that the field variable V(r, t) and the source variable Q(r, t) are coupled by the inhomogeneous wave equation (2.1), the cross-spectral densities Wo(0) and w(O) are also related. This relationship is rather complicated. However, it has been shown in ref. [1] that there exists a relatively simple relationship between the four-dimensional spatial Fourier transforms
X exp {-i(1'l . r 1 +1"2 "r2)} d2r 1 d2r2 , t i~(O)( f f2,60) _ 1 ~'~1' - (2~
(2.6)
f f w ( O ) ( r l , r2,60)
X exp {-i(fl • r I +f2 "r2)} d2rl d2r2
(2.7)
of these quantities. Here f l and f2 are each a two-dimen sional spatial frequency variable and the integrations on the right-hand-sides of eqs. (2.6) and (2.7) are taken twice independently over source plane z = 0. The relationship between I~o(0) and I~a(°) established in ref. [1] is w(O)t/" ,f2,60 ) = (2n) 2 ~
-t
glg2
)f)_1,.,,60_,
(2.8)
where when//2 < k 2 (2.9) =bv/~2 - k 2
when f/2 > k 2,
(j = 1,2) with
(2.10)
k -- 60/c,
It has also been shown in ref. [1] that the spatial Fourier transforms Wv(0) and ~(0) of the cross-spectral density functions wv(O)and w(O~ have another significance, being trivially related to the cross-spectral density functions of the two-dimensional spatial Fourier transforms 1 fu(O)(r, co) e-i/. r d2r, 7(O)(f, 60) = (2n.)2
;(o)¢, 60) =
(2rrF
f p ( % , 60) e-if., d2r,
(2.11)
(2.12)
of v(O)(r, 60) and o(O)(r, 60) respectively. These crossspectral densities are defined by the formulas (o'(O)(fl , co)~'(O)* (f2 , CO')) = w(O)(f 1 '1'2' 60) ~(60--co')' (2.13)
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June 1978
(~(O)(fl, to)p'~(O)*oe2, CO')>= w(O)oeI ' 6 ' to) ~5(to--to')
0 < I/a(°)ffl,[2, to)l < l,
(2.14)
O < ~,~, , j l , f 2 , t o ) [ <---1,
and the above mentioned relationship established in ref. [ 1] is w~O)(fl ' f 2 ' to) = [4"(0)//'ff ~ l ' --f2' ¢0),
(2.15)
w(O)(fl '[2' to) = w(O)Otl'-[2' to).
(2.16)
The formulas (2.15) and (2.16) may be regarded as analogous to the well known Wiener-Khintchine theorem (e.g. [4] ) in the space-frequency (rather than the timefrequency) domain. If we make use of eqs. (2.15) and (2.16) the relation (2.8) is seen to imply that (2rr) 2
w(O)t¢ 'f2' to) = w(O)t¢ r co). u" Ul gig ~ b" ~ l ' J 2 '
(2.17)
, (o)~f
(3.3) (3.4)
for all values of their arguments. The extreme values unity and zero of ~uo~)(fl,f2, co) indicate complete correlation and complete absence of correlation respectivel~ between the spatial frequency components ~'(0)(f 1,to) and b"(0)(f 2, to) of the field variable at frequency co in the source plane. The extreme values unity and zero °f/a(0)0"1,1"2, to)have the same significance with respecl to t~e spatial frequency components ~'(0)(f, co) of the source variable. If we substitute on the right-hand-side of eq. (3.1) for WoD)from the relation (2.17) and make use of (3.2) we find after a straightforward calculation that the two correlation coefficients are related as follows: , ~" (O)(f ~ai ,f2, to) = Oil2/'/(O)(fl ' f 2 ' to),
(3.5)
where
3. Correlation coefficients in the source plane We will now show that an interesting consequence of the formula (2.8) is revealed when one considers certain correlation coefficients involving the spatial Fourier transforms of the field variable u(°)(r,to) and of the source variable p(°)(r, co). We define these correlation coefficients by the formulas
6, to) /t~)(/1 f2 to)
rl
)
22
to) (3.1)
Ot12 = 1
when]'2 < k 2 andf~ < k 2 }
(3.7a)
or when f~l > k2 and [ i > k2 =i
when ]'12 < k 2 and [~ > k 2
(3.7b)
=-i
w h e n f ~ > k 2 a n d . f i < k 2.
(3.7c)
If we express the two correlation coefficients in the form
u~o)O'l,h, to) - tao~),~ U l ' [2' co)lexp (i~p~)(fl, f2' co)),
and
to)-
(3.8)
w~)(¢1,I2, to)
,,/w(o)l¢ ,~ w~/w(o)~r r to)' v, ~ ~J1,J1, / v, ~' '~2,J2, (3.2)
where WoL°)0"I,f2' 60) and Wtg)r¢,o IJ1, I2 ~ , to) are the crossspectral density functions defined by eqs. (2.13)and (2.14) respectively. In a strictly similar manner as was demonstrated in connection with another correlation coefficient in ref. [3] it is not difficult to show that the correlation coefficients defined by eqs. (3.1) and (3.2) are bounded by zero and unity in absolute values, i.e., that 290
/----g /-----~, . c~12 = vglglVg2g21glg 2. (3.6) Making use of the explicit expressions (2.9) for the quantitiesg 1 and g2 it follows that
, (0)~ ,[2, to) = lu~)q'l , h ' to ) lex P ~",' ~~0)~r ,,1 f2, to)), (3.9) the relation (3.5), together with eqs. (3.7), implies that for all values of the arguments
lULo) o (fl ,[2, to)[ = lu~)O'l ,.t"2 , to) I and that
(3.1o)
Volume 25, number 3
when/12 < k 2 and ].2 < k 2 }
June 1978
OPTICS COMMUNICATIONS
(3.1 la)
or when ].12> k 2 and].~ > k 2 = ~°~)(fl' ].2' to) + n/2
when ].12< k 2 and ].~ > k 2
(3.1 lb)
S°urcI I I~~ ' Cj-v~'~~ '
~
IO z
= ¢~9)(fl '].2' to) - n/2
when ].12> k 2 and ].22 < k 2.
(3.1 lc) Fig. I. Illustratingthe notation relating to eq. (4.1).
Tlae formula 0.5), together with eqs. (3.?) or, equivalently, the more explicit formulas (3.10) and (3.11) are one of the main results of this note. They bring into evidence a remarkably simple relationship that exists between the correlations in the field distribution and in the source distribution in the plane of the source, when each is expressed in terms of the frequency variables (]., co) rather than in terms of the space-time variables (r, t).
4. The degree of spatial coherence of the far field and its relationships to the correlation coefficients in the source plane
Another interesting and simple relationship is revealed when one considers the correlation coefficient (the complex degree of spatial coherence) of the far field generated by the source. To show this we first recall that the cross-spectral density function of the field at points P1 and P2 in the far zone, at distances R 1 and R 2 from the origin, in directions specified by unit vectors Sl and s 2 respectively (see fig. 1) may be expressed in the form [ref. 1, eq. (3.5)] W~=)(R 151 , R2S2, 6o) (4.1) = (2Tr)4~(0)(k$1 ±' _k$2± ' to)eik(g l-g 2)/R 1R2. Here ~(0) is, of course, the four-dimensional spatial Fourier°transform, defined b~ eq. (2.7), of the crossspectral density function Wp(0) of the source distribution at frequency 6o and $1± and s2± are the two-dimensional vectors obtained by projecting the three-dimensional unit vectors Sl and $2 onto the source plane z =
0. The formula (4.1), which is to be interpreted in the asymptotic sense (indicated by superscript co) as kR 1 -, oo and kR 2 -* 0% may be expressed in the following alternative form by use of the relation (2.16): W(~*)(R 151 , R2S 2 , co) = (2n)4leff)(kSli, k$2±' to)eik(R t -R2 )/R 1R2 . (4.2) Let us now introduce the degree of spatial coherenc~ [ref. 3~ eq. (2.10)] of the light oscillations at the points P1 and P2, viz. i.t(~)(R ~ ' R2s2, to) o ~ 1~1 (**) Wv (RI$1,R2$2,to)
(4.3)
D
X/'W~)(R lsl ,R lSl, to) x/W(°°)(R2s2,R2s2 , co) On substituting from (4.2) on the r.h.s, of (4.3) we obtain the following expression for/%(~): ttv(~) (RlS 1 , R2s2, co) (4.4) 14A~)(kSlt, ks2±, to)
kSsi, to)
elk(R 1-R:
k$a, to)
The first factor on the r.h.s, of (4.4) is, according to (3.2), precisely the degree of correlation #~)(kSl± , ks2.t co). Moreover, the spatial frequency arguments kSl± = f l , k$2± =f2 are each necessarily smaller than k in abso. lute values (i.e., I/'11< k, If21 < k) because Sl± and s2± are projections of unit vectors. Hence with these arguments the condition of the first line in eq. (3.7a) applie: 291
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OPTICS COMMUNICATIONS
so that ~12 = 1 and we obtain at once from (4.4), (3.2) and (3.5) the interesting double relation /l v(**)R(lSl , R2s2,co)=ll~)(kSll, ks2±,co)e ik(R~-R2) (4.5a) =/a,~)(kSl±, ks2± ' co)elk (R i -R2). (4.5b) In particular, if the points P1 and P2 in the far zone are at equal distances (R l = R 2 = R) from the origin, eqs. (4.5) reduce to /a!'~)(R lSl , R282, co) =/a~)(k$1±, ks2± , ~ ) = tl-~)(kSli, ks2±, co).
June 1978
was derived with the help of the formula (3.5) involving/a~ ), it is valid irrespective of whether the field is generated by a primary or by a secondary source (in the latter c a s e / ~ ) is not defined). That the relation holds also for a secondary source can readily be shown by taking as a starting point of the derivation the following formula [5] in place o f e q . (4.1):
W!~)(RlS 1 , R2s 2, w) = (27rk)2cos 01 cos 02 X W~0)(k$1±, -ks2£ , w)e ik(R 1-R2)/R IR2 ,
(4.7)
where 01 and 02 are the angles which the unit vectors s I and s 2 make with the normal to the plane z = 0.
(4.6)
References Thus the degree of spatial coherence p(**) of the field at any two points in the far zone at equal distances from the origin is equal to either of the two correlation coefficients ta~ ) and/a~ ) of the spatial frequency components of the source distribution and of the field distribution, with the arguments shown explicitly in eq. (4.6). Although the relation (4.5b) between U(o~*) and/a~ ),
292
[1 } E. Wolf and W.ll. Carter, J. Opt. Soc. Amer., in press. [2] M. Born and E. Wolf, Principles of optics, 5th Ed. (Pergamon Press, New York, 1975), Sec. 10.2. [31 L. Mandel and E. Wolf, J. Opt. Soc. Amer. 66 (1976) 529, eq. (2.4). [4] C. Kittel, Elementary statistical physics (J. Wiley, New York, 1958) p. 133 et seq. [5] E.W. Marchand and E. Wolf, J. Opt. Soc. Amer. 62 (1972) 379, eq. (34).
OPTICS COMMUNICATIONS
June 1978
SOME RELATIONSHIPS BETWEEN THE CORRELATION COEFFICIENTS OF PLANAR SOURCES AND OF THEIR FAR FIELDS William H. CARTER
Naval Research Laboratory, Washington, D.C. 20375, USA and Emil WOLF
Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, N.Y. 14627, USA Received 8 February 1978 Two new correlation coefficients are introduced, that characterize the correlations between tbe spatial frequency components of the ,source distribution and of the field distribution respectively, in the plane of a two-dimensional source. It is then shown that except possibly for a simple phase factor depending on the magnitudes of the .spatial frequencies, the two correlation coefficients are equal to each other. It is further shown that the degree of spatial coherence of the far field generated by the ,source is equal, except possibly for a simple geometrical phase factor, to either of the two new correlation coefficients relating to the distributions in the source plane.
1. Introduction In a recent paper [1] a certain relationship was established between the cross-spectral density functions of the source distribution and of the field distribution in the plane of a primary two-dimensional source. In the present paper we show, with the help of this result, that there exists a simple relationship between certain new correlation coefficients that characterize the correlations between the spatial frequency components of the source distribution and of the field distribution respectively in the plane of the source. In fact, the two correlation coefficients are found to be equal, except possibly for a simple phase factor depending on the magnitudes of the spatial frequencies. We also show that the degree of spatial coherence of the far field generated by the source is equal, except possibly for a simple phase factor depending on geometry, to either of the two new correlation coefficients relating to the distributions in the source plane.
2. Field correlation functions and source correlation functions We consider an optical field V(r, t), generated by a primary source distribution Q(r, t) in free space. Both V(r, t) and Q(r, t) are taken to be scalar functions of position (r) and time (t). We assume that the source distribution Q(r, t) is localized, for all time, in some finite domain around the origin r = 0. We take V(r, t) and Q(r, t) to be analytic signals :[: and we consider them to be stationary random variables. They are coupled by the inhomogeneous wave equation V2 V(r, t) - 1
~2 V(r, t) . . . . 4~Q(r, t), c 2 ~t 2
(2.1)
where c is the vacuum speed of light. Let
1
v(r, ~ ) = ~ f
V(r,t)e it°t dt,
(2.2)
$ For the definition of analytic signals see, for example, Born and Wolf {21. 288
Volume 25, number 3
OPTICS COMMUNICATIONS
and
,y Q(r,t)e i°~t dt
p(r, co) = ~
~(0)¢~ l ffw~O)(rl,r2,60) o 'J1'1'2' 60) = (2n)4 (2.3)
--oo
be the Fourier transforms of V(r,t) and of Q(r,t) respectively. As is well known from the theory of stationary random processes the Fourier transforms (2.2) and (2.3) do not exist as ordinary integrals and must be interpreted in the sense of the theory of generalized functions. Let us now introduce the cross-spectral density functions Wo(r1,r2, 6o) and Wa(rl, r2, co) of the field- and the source variable respectively, via the relations (e.g. 13] )
(v(rl, 60)v*(r2,60')) = Wv(rl, r2,60)5(w - co'),
June 1978
(2.4)
(p(r a,co)p*(r2,w' )) = Wo(r 1 , r 2 , w ) 5 ( c o - co'). (2.5) The Dirac delta function 5 appears on the right-handsides of these equations as consequence of the assumed stationarity of the field and of the source. It is clear from the above formulae that Wv(rl, r2, w) represents the correlation that exists between the (generalized) Fourier components of frequency 60 of the light oscillations at points r 1 and r 2 in the field. Wo(rl, r2, 60) has, of course, a similar significance with respect to the source oscillations. Suppose now that the source is two-dimensional, occupying a finite portion of the plane z = 0 and that it radiates into the half-space z > 0. We will indicate by a superscript zero values of quantities pertaining to the source plane. Thus, for example u(0)(rl, w) indicates the values of o(rl, 60) when the point specified by the position vector r 1 is located in the source plane z = O. W(°)(rl, rE, 60) indicates the value of Wo(rl, r2, co) when both the points, specified by the position vectorsr 1 and r2, are located in this plane, etc. Now as a consequence of the fact that the field variable V(r, t) and the source variable Q(r, t) are coupled by the inhomogeneous wave equation (2.1), the cross-spectral densities Wo(0) and w(O) are also related. This relationship is rather complicated. However, it has been shown in ref. [1] that there exists a relatively simple relationship between the four-dimensional spatial Fourier transforms
X exp {-i(1'l . r 1 +1"2 "r2)} d2r 1 d2r2 , t i~(O)( f f2,60) _ 1 ~'~1' - (2~
(2.6)
f f w ( O ) ( r l , r2,60)
X exp {-i(fl • r I +f2 "r2)} d2rl d2r2
(2.7)
of these quantities. Here f l and f2 are each a two-dimen sional spatial frequency variable and the integrations on the right-hand-sides of eqs. (2.6) and (2.7) are taken twice independently over source plane z = 0. The relationship between I~o(0) and I~a(°) established in ref. [1] is w(O)t/" ,f2,60 ) = (2n) 2 ~
-t
glg2
)f)_1,.,,60_,
(2.8)
where when//2 < k 2 (2.9) =bv/~2 - k 2
when f/2 > k 2,
(j = 1,2) with
(2.10)
k -- 60/c,
It has also been shown in ref. [1] that the spatial Fourier transforms Wv(0) and ~(0) of the cross-spectral density functions wv(O)and w(O~ have another significance, being trivially related to the cross-spectral density functions of the two-dimensional spatial Fourier transforms 1 fu(O)(r, co) e-i/. r d2r, 7(O)(f, 60) = (2n.)2
;(o)¢, 60) =
(2rrF
f p ( % , 60) e-if., d2r,
(2.11)
(2.12)
of v(O)(r, 60) and o(O)(r, 60) respectively. These crossspectral densities are defined by the formulas (o'(O)(fl , co)~'(O)* (f2 , CO')) = w(O)(f 1 '1'2' 60) ~(60--co')' (2.13)
289
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June 1978
(~(O)(fl, to)p'~(O)*oe2, CO')>= w(O)oeI ' 6 ' to) ~5(to--to')
0 < I/a(°)ffl,[2, to)l < l,
(2.14)
O < ~,~, , j l , f 2 , t o ) [ <---1,
and the above mentioned relationship established in ref. [ 1] is w~O)(fl ' f 2 ' to) = [4"(0)//'ff ~ l ' --f2' ¢0),
(2.15)
w(O)(fl '[2' to) = w(O)Otl'-[2' to).
(2.16)
The formulas (2.15) and (2.16) may be regarded as analogous to the well known Wiener-Khintchine theorem (e.g. [4] ) in the space-frequency (rather than the timefrequency) domain. If we make use of eqs. (2.15) and (2.16) the relation (2.8) is seen to imply that (2rr) 2
w(O)t¢ 'f2' to) = w(O)t¢ r co). u" Ul gig ~ b" ~ l ' J 2 '
(2.17)
, (o)~f
(3.3) (3.4)
for all values of their arguments. The extreme values unity and zero of ~uo~)(fl,f2, co) indicate complete correlation and complete absence of correlation respectivel~ between the spatial frequency components ~'(0)(f 1,to) and b"(0)(f 2, to) of the field variable at frequency co in the source plane. The extreme values unity and zero °f/a(0)0"1,1"2, to)have the same significance with respecl to t~e spatial frequency components ~'(0)(f, co) of the source variable. If we substitute on the right-hand-side of eq. (3.1) for WoD)from the relation (2.17) and make use of (3.2) we find after a straightforward calculation that the two correlation coefficients are related as follows: , ~" (O)(f ~ai ,f2, to) = Oil2/'/(O)(fl ' f 2 ' to),
(3.5)
where
3. Correlation coefficients in the source plane We will now show that an interesting consequence of the formula (2.8) is revealed when one considers certain correlation coefficients involving the spatial Fourier transforms of the field variable u(°)(r,to) and of the source variable p(°)(r, co). We define these correlation coefficients by the formulas
6, to) /t~)(/1 f2 to)
rl
)
22
to) (3.1)
Ot12 = 1
when]'2 < k 2 andf~ < k 2 }
(3.7a)
or when f~l > k2 and [ i > k2 =i
when ]'12 < k 2 and [~ > k 2
(3.7b)
=-i
w h e n f ~ > k 2 a n d . f i < k 2.
(3.7c)
If we express the two correlation coefficients in the form
u~o)O'l,h, to) - tao~),~ U l ' [2' co)lexp (i~p~)(fl, f2' co)),
and
to)-
(3.8)
w~)(¢1,I2, to)
,,/w(o)l¢ ,~ w~/w(o)~r r to)' v, ~ ~J1,J1, / v, ~' '~2,J2, (3.2)
where WoL°)0"I,f2' 60) and Wtg)r¢,o IJ1, I2 ~ , to) are the crossspectral density functions defined by eqs. (2.13)and (2.14) respectively. In a strictly similar manner as was demonstrated in connection with another correlation coefficient in ref. [3] it is not difficult to show that the correlation coefficients defined by eqs. (3.1) and (3.2) are bounded by zero and unity in absolute values, i.e., that 290
/----g /-----~, . c~12 = vglglVg2g21glg 2. (3.6) Making use of the explicit expressions (2.9) for the quantitiesg 1 and g2 it follows that
, (0)~ ,[2, to) = lu~)q'l , h ' to ) lex P ~",' ~~0)~r ,,1 f2, to)), (3.9) the relation (3.5), together with eqs. (3.7), implies that for all values of the arguments
lULo) o (fl ,[2, to)[ = lu~)O'l ,.t"2 , to) I and that
(3.1o)
Volume 25, number 3
when/12 < k 2 and ].2 < k 2 }
June 1978
OPTICS COMMUNICATIONS
(3.1 la)
or when ].12> k 2 and].~ > k 2 = ~°~)(fl' ].2' to) + n/2
when ].12< k 2 and ].~ > k 2
(3.1 lb)
S°urcI I I~~ ' Cj-v~'~~ '
~
IO z
= ¢~9)(fl '].2' to) - n/2
when ].12> k 2 and ].22 < k 2.
(3.1 lc) Fig. I. Illustratingthe notation relating to eq. (4.1).
Tlae formula 0.5), together with eqs. (3.?) or, equivalently, the more explicit formulas (3.10) and (3.11) are one of the main results of this note. They bring into evidence a remarkably simple relationship that exists between the correlations in the field distribution and in the source distribution in the plane of the source, when each is expressed in terms of the frequency variables (]., co) rather than in terms of the space-time variables (r, t).
4. The degree of spatial coherence of the far field and its relationships to the correlation coefficients in the source plane
Another interesting and simple relationship is revealed when one considers the correlation coefficient (the complex degree of spatial coherence) of the far field generated by the source. To show this we first recall that the cross-spectral density function of the field at points P1 and P2 in the far zone, at distances R 1 and R 2 from the origin, in directions specified by unit vectors Sl and s 2 respectively (see fig. 1) may be expressed in the form [ref. 1, eq. (3.5)] W~=)(R 151 , R2S2, 6o) (4.1) = (2Tr)4~(0)(k$1 ±' _k$2± ' to)eik(g l-g 2)/R 1R2. Here ~(0) is, of course, the four-dimensional spatial Fourier°transform, defined b~ eq. (2.7), of the crossspectral density function Wp(0) of the source distribution at frequency 6o and $1± and s2± are the two-dimensional vectors obtained by projecting the three-dimensional unit vectors Sl and $2 onto the source plane z =
0. The formula (4.1), which is to be interpreted in the asymptotic sense (indicated by superscript co) as kR 1 -, oo and kR 2 -* 0% may be expressed in the following alternative form by use of the relation (2.16): W(~*)(R 151 , R2S 2 , co) = (2n)4leff)(kSli, k$2±' to)eik(R t -R2 )/R 1R2 . (4.2) Let us now introduce the degree of spatial coherenc~ [ref. 3~ eq. (2.10)] of the light oscillations at the points P1 and P2, viz. i.t(~)(R ~ ' R2s2, to) o ~ 1~1 (**) Wv (RI$1,R2$2,to)
(4.3)
D
X/'W~)(R lsl ,R lSl, to) x/W(°°)(R2s2,R2s2 , co) On substituting from (4.2) on the r.h.s, of (4.3) we obtain the following expression for/%(~): ttv(~) (RlS 1 , R2s2, co) (4.4) 14A~)(kSlt, ks2±, to)
kSsi, to)
elk(R 1-R:
k$a, to)
The first factor on the r.h.s, of (4.4) is, according to (3.2), precisely the degree of correlation #~)(kSl± , ks2.t co). Moreover, the spatial frequency arguments kSl± = f l , k$2± =f2 are each necessarily smaller than k in abso. lute values (i.e., I/'11< k, If21 < k) because Sl± and s2± are projections of unit vectors. Hence with these arguments the condition of the first line in eq. (3.7a) applie: 291
Volume 25, number 3
OPTICS COMMUNICATIONS
so that ~12 = 1 and we obtain at once from (4.4), (3.2) and (3.5) the interesting double relation /l v(**)R(lSl , R2s2,co)=ll~)(kSll, ks2±,co)e ik(R~-R2) (4.5a) =/a,~)(kSl±, ks2± ' co)elk (R i -R2). (4.5b) In particular, if the points P1 and P2 in the far zone are at equal distances (R l = R 2 = R) from the origin, eqs. (4.5) reduce to /a!'~)(R lSl , R282, co) =/a~)(k$1±, ks2± , ~ ) = tl-~)(kSli, ks2±, co).
June 1978
was derived with the help of the formula (3.5) involving/a~ ), it is valid irrespective of whether the field is generated by a primary or by a secondary source (in the latter c a s e / ~ ) is not defined). That the relation holds also for a secondary source can readily be shown by taking as a starting point of the derivation the following formula [5] in place o f e q . (4.1):
W!~)(RlS 1 , R2s 2, w) = (27rk)2cos 01 cos 02 X W~0)(k$1±, -ks2£ , w)e ik(R 1-R2)/R IR2 ,
(4.7)
where 01 and 02 are the angles which the unit vectors s I and s 2 make with the normal to the plane z = 0.
(4.6)
References Thus the degree of spatial coherence p(**) of the field at any two points in the far zone at equal distances from the origin is equal to either of the two correlation coefficients ta~ ) and/a~ ) of the spatial frequency components of the source distribution and of the field distribution, with the arguments shown explicitly in eq. (4.6). Although the relation (4.5b) between U(o~*) and/a~ ),
292
[1 } E. Wolf and W.ll. Carter, J. Opt. Soc. Amer., in press. [2] M. Born and E. Wolf, Principles of optics, 5th Ed. (Pergamon Press, New York, 1975), Sec. 10.2. [31 L. Mandel and E. Wolf, J. Opt. Soc. Amer. 66 (1976) 529, eq. (2.4). [4] C. Kittel, Elementary statistical physics (J. Wiley, New York, 1958) p. 133 et seq. [5] E.W. Marchand and E. Wolf, J. Opt. Soc. Amer. 62 (1972) 379, eq. (34).