- Email: [email protected]
Completely coherent radiation
.\NX.~LS
OF
PHYSICS:
27, 72-78
(1964)
Completely
Coherent WILLIAM
Department
of Electrical
Engineering,
Radiation*
STREIFEIZ University
of Rochester,
Rochester,
AVew l-ark
Some aspects of the scalar coherence theory of radiation employing two-point correlation functions are considered. In particular, it is shown that “real” waveforms in polychromatic fields need not, appear similar to be considered coherent under the present definition. The decomposition of t,he mutual coherence function for such fields is discussed. ITnder certain assumptions a general form for coherent fields is derived and these fields are shown to satisfy the equations of geometric opt,ics. For homogeneous media the surfaces of constant phase are the so-called cyclides of I>upin. The significance of these results in relating coherent, sources and fields is mentioned as is t,he decomposition of incoherent sources. I. INTROL)IlCTIOD;
Completely coherent radiation has been examined during the past few years by Mandel and Wolf (1) and Parrent (2, S), but several questions in this area are as yet unresolved. These concern the extension of the theory to include polychromatic radiation. Xew results relating to completely coherent fields are derived. II.
BASIC
REfiUI,T,‘:
As in ref. 1 we represent the radiation disturbance (P, t) in a stationary scalar field by a complex analytic coherence function is defined by T Ji(i T
+ T)Vz*(t)
at the space time point signal V(f). The mutual
dt = (V,(t
+ T)v?*(~)),
(1)
where VI and Vz are the disturbances at PI and Pf respectively, and the degree of coherence is expressed in terms of the normalized mutual coherence function as follows : y1*( 7) = rl*(T)/l:‘*I;‘*. Here II = m(O) * This
work
(2)
and I? = I’?*(O). The disturbances
was supported
hg the
National
Science 72
VI and I72 are said to be
Foundation.
CIOMPLETELT
completely
coherent
COHEKENT
ItAL)I.~TIOS
73
(1) if there exists one value T” of 7 such that / Yd7”)l
= 1.
Often the analysis of coherence is limited mean frequency F, and t’he approximation - ylz( ~(,)p~?nG(‘,l’~ii,), YE ( 70’ )
(3)
to quasimonochromatic
radiation
of
17,; - 7111 << 1 ~‘Av, (4)
is employed, where AV is the effective bandwidth of the radiation. Then if (3) is satisfied, one can always define T,,’ such that yl?(~“‘) = 1. There is then little or no distinction between defining coherence in terms of the modulus or the real part of yl!( T). This is not true for polychromatic radiation, and it is shown below that (3) permits situat,ions which may not be physically satisfactory. Consider the following two cases : ( 1 ), let the real disturbances at I’, and I’, he I.(,f ). The corresponding analytic signals are l,(f) where s(t)
is the Hilbert
= l’,(f)
transform
= I.(f) + is(f),
of l,(t).’
Then
so that the disturbances are coherent. (2), retain r,(t) at I’, , but substitute s(t) at I’, . These functions are different; however, r-,(t)
= r(f)
+ is(f)
T’,(f)
=
-
and s(f)
G(f)
so that
Thus according to (8) these disturbances are coherent. L4t optical and microwave ( > 10” cps) frequencies it is possible that “real” waveforms such as i.(f) above cannot be observed. However, at lower frequencies ( < 10’ cps) “real” waveforms may be observed on oscilloscopes. Thus one has the seemingly anomalous situation in which two waveforms appear different, but are coherent. We must therefore distinguish between coherence and similarity. It should be noted that a *:‘3 phase shift of all Fourier components converts nil) to s(f) and so the 1 For stationary fields the integral defining the conjugate field may diverge. To avoid this difficulty one may first assume that the field exists only for a finite time int,crval - 7’ 5 ts 7 and one can proceed to the limit I‘I--) m at the end of the calcnlat,ions.
74
STKE
I FE I<
definition (3) includes similarity if we permit devices which shift all relevant frequencies equally. It is indeed possible to construct (at least at, low enough frequencies) an electric filter which approximately shifts all frequencies in a finite band by a/2 (4). In the following we will continue to employ (3) as the definition of coherence. Eyuation (3) may he rewritten in the form, c.f. ref. 1,
(5) and by defining
one has the conditions (7) and (I’l(t III.
,2 completely
+ 7)15’(r))
= (I’dt
COMPLETEI,\i
+ T)Ir-(t))
COHEREST
= 0.
(8)
FIELI)S
coherent field for a region R is one in which 1 Ytj(Tij)J
= 1
for all pairs of points Pi and Pj in R for a least OIW value of 7ij . This is the definition given by 1Iandel and Wolf ( I ), but does not agree with Parrent’s (3). He requires / y12( T) ( = 1 for all T, which is perhaps too restrictive since it appears to admit only monochromatic fields. It is convenient to employ a reference point P, and write 7i0 = 7;) so that = Ti It is generally accepted (I, 2) that a completely coherent quasimonochromatic field permits the decomposition
Tij
Tj.
where
However, since I’iJ(T
71j
+ 2?TiL) =
yij(
Tij)
from (9) that one may select = I.
COMPLETELY
COHERENT
I~.-\I~I.~TION
Then Eq. (6) enables us to mrit,e l’,(f
+ ZkT;)
= I’,(f)
+ IV”(f),
where (IId t)rr;,*(t),
= 0,
and q’,(~f
It follows
+
7)11’“*(f))
=
0.
then, that Y,,(T)
= Y!,(T + ST/b)
and yI,( T) is periodic and monochromatic, i.e., it has a single d-function spectrum at 8. Thus the decomposition (9) holds only for strictly monochromatic radiation, but the mutual coherence function factors in the form ( 10) for all fully coherent radiation. This is because all such radiation exhibits cross-spectral purity ( 5), as is shown below. We write
and employ
( 6). One then finds
and r,j(T +
Tsj)
= It”It”ytj(
Tzj)yjj(
T).
(10)
The use of ( 9) has in fact assumed that y,,(T) = e-“+, In the foregoing, the function II’( 0, defined by (6), was employed. It would be very convenient to have It’(t) = 0, and indeed this will be assumed in the remainder of the work. To justify this assumption we note that W(t) is a realization of a process whose mean and mean square values are both zero, cf. EC{. (i). Thus, in the duration 7’, the lengt’h of time, ?IP , for which 1 Il’if)l
> E
is such that lim (7’P( e) /T) T-+% for any E > 0.
= 0,
Consider now a completely coherent field whose realization point is the analytic signal j(f). Since according to the above the disturbance at every other point must have the same time possible shift and complex proportionality factor, the most field may be expressed in the form G(.r, y, 2, f) = 11(x, y, z),f[f -
at some reference made assumption variation, with a general coherent
11(x, y, 2) “Cl
(.ll)
where
in a source free region. It has been assumed that all functions are differentiable as required and the index of refraction, n, depends only on position. The result of writing f( t - U/C) = j( [) and substituting (11 ) into ( 12) is
ffh Equation lut,ion
- ; f (I&
+ 2Vh.Vu) + $ Jf& (vu .VlL - ,?a’)= 0.
( 13) has a large class of solutions .f(<) = j(f)
= Ae’“’
including
the monochromatic
(13) so-
+ Be?,
which is obtained by setting u(x, y, Z) = 0. Of somewhat greater interest are those cases in which .f( <) is not determined by (13), thus permitting arbitrary source fluctuations. Then ( 18) must be satisfied identically by requiring that the coefficients of .f, r/,f!d<, and kff!dS,’ vanish individually, i.e., hiVu.Vu
- n”) = 0,
hV2u + 2VU’Vh
= 0,
( I-la) (l-lb)
and v’h = 0.
(1Jc)
It is interesting to note that these equations are identical with the equations of geometric optics. Physically this occurs because arbitrary source fluctuations permit discontinuities in the field, which propagate according to the laws of geometric optics (6). Friedlander (7) has considered systems of equations such as (Ma)-( 14~) with n constant, say n = I. He has shown that the most general solutions for the u = const. surfaces are the cyclides of Dupin (8), which include, the plane, sphere, cylinder, cone, torus, etc. 1Iost of these surfaces require that h(.c, y, z) be multivalued (7), but G(x, y, z, f) must be single valued. Thus for these to be
admissible solutions an obstacle must Icor example, nh~~ u = 1’ in cylindrical
be present to assure siriglc-valuedness. coordinates
and a screen must wstrict thr range of 0 t’o less In the cast of a finit’c volume of so~uws in solutions are allowed in the cstJerior region and cyclides, planes, and spheres. The first of these “c(2 + y? + 2”) + (2 + /j)( I -
than 2~. empty space only single valued onr is then limited to parabolic is defitwd hy ( 9)
IL) - z’( 1 + u) -
(.r - 1 -
(15) 11) (~1+ Il.)- = 0,
where 1 is an arbitrary cwlstjant. l“or large values of ZLtltrsc surfaws do not approach spheres or equivalcutly the Sommerfeld radiation condition is not satisfied, i.e., these solutions arc not permissible in our problem. l’lane waves are similarly eliminated, but we note in passing that the solution is then h(r,
y),f(i - z/c),
where C’h.(.r, 7J) = 0. Finally,
spherical waves are expressible
in the form (1Ga)
r
sin e fj
sin e $ (
>
”
+ i$ = 0. (0
(1Gh i
However, the only finite solut’ion of i IGh) at 0 = 0 is w = const., so that the only coherent field produced hy randomly fluctuating sources in a finite volume of empt#y space is the spherical wave
; 82 - I./C),
(17)
which includes the plane wave as a limiting case. If a coherent source is defined as one which produces a coherent field throughout the source region, then it is evident that the fields in the source free region need not, he fully coherent. Furthermore the decomposition of an incoherent source into infinitesimal sources producing coherent fields is equivalentj to stating that these sources produce waves of t’he form ( 17).
78
STREIFEK .kXNoWLEDGMENT
The author wishes to thank Drs. E. Wolf Iat)ing to various aspects of this work.
RECEIVED:
July
15,
and D. S. Ruchkiu
for several
discussions
re.
1963 REFERENCES
1, I,. M.INDEL .GD E. WOI~F, J. Opt. Sot. :lm. 51, 815 (1961). 2. (i. B. PARRENT, J. Opt. Sot. ilm. 49, 787 (1959). 3. G. B. PARRENT, Opt. .4cta 6, 285 (1959). 4. J. E. STORER. “Passive Network Synthesis,” p. 298. McGraw--Hill, Sew York, 1957. 5. L. MANUEL, J. Opt. Sot. Am. 51, 1342 (1961). 6. M. BORN AND E. WOLF, “Principles of Optics,” p. 108 aud Appendix \I. Pergaruon: London, 1959. i'. F. (i. FRIEDL.INDER, Proc. Vumb~itlge Phil. Sot. 43, 366 (1947). 8. A. R. FORSYTH “I)ifferentinl Geometry,” p. 324, Cambridge ITniv. Press, 1912; or A. C.\YLEY, “Maihematical Papers,” vol. IS, p. 64. Cambridge Ilniv. Press, 1896. 9. A. R. FORSYTH, lot. cit. p. 328.
OF
PHYSICS:
27, 72-78
(1964)
Completely
Coherent WILLIAM
Department
of Electrical
Engineering,
Radiation*
STREIFEIZ University
of Rochester,
Rochester,
AVew l-ark
Some aspects of the scalar coherence theory of radiation employing two-point correlation functions are considered. In particular, it is shown that “real” waveforms in polychromatic fields need not, appear similar to be considered coherent under the present definition. The decomposition of t,he mutual coherence function for such fields is discussed. ITnder certain assumptions a general form for coherent fields is derived and these fields are shown to satisfy the equations of geometric opt,ics. For homogeneous media the surfaces of constant phase are the so-called cyclides of I>upin. The significance of these results in relating coherent, sources and fields is mentioned as is t,he decomposition of incoherent sources. I. INTROL)IlCTIOD;
Completely coherent radiation has been examined during the past few years by Mandel and Wolf (1) and Parrent (2, S), but several questions in this area are as yet unresolved. These concern the extension of the theory to include polychromatic radiation. Xew results relating to completely coherent fields are derived. II.
BASIC
REfiUI,T,‘:
As in ref. 1 we represent the radiation disturbance (P, t) in a stationary scalar field by a complex analytic coherence function is defined by T Ji(i T
+ T)Vz*(t)
at the space time point signal V(f). The mutual
dt = (V,(t
+ T)v?*(~)),
(1)
where VI and Vz are the disturbances at PI and Pf respectively, and the degree of coherence is expressed in terms of the normalized mutual coherence function as follows : y1*( 7) = rl*(T)/l:‘*I;‘*. Here II = m(O) * This
work
(2)
and I? = I’?*(O). The disturbances
was supported
hg the
National
Science 72
VI and I72 are said to be
Foundation.
CIOMPLETELT
completely
coherent
COHEKENT
ItAL)I.~TIOS
73
(1) if there exists one value T” of 7 such that / Yd7”)l
= 1.
Often the analysis of coherence is limited mean frequency F, and t’he approximation - ylz( ~(,)p~?nG(‘,l’~ii,), YE ( 70’ )
(3)
to quasimonochromatic
radiation
of
17,; - 7111 << 1 ~‘Av, (4)
is employed, where AV is the effective bandwidth of the radiation. Then if (3) is satisfied, one can always define T,,’ such that yl?(~“‘) = 1. There is then little or no distinction between defining coherence in terms of the modulus or the real part of yl!( T). This is not true for polychromatic radiation, and it is shown below that (3) permits situat,ions which may not be physically satisfactory. Consider the following two cases : ( 1 ), let the real disturbances at I’, and I’, he I.(,f ). The corresponding analytic signals are l,(f) where s(t)
is the Hilbert
= l’,(f)
transform
= I.(f) + is(f),
of l,(t).’
Then
so that the disturbances are coherent. (2), retain r,(t) at I’, , but substitute s(t) at I’, . These functions are different; however, r-,(t)
= r(f)
+ is(f)
T’,(f)
=
-
and s(f)
G(f)
so that
Thus according to (8) these disturbances are coherent. L4t optical and microwave ( > 10” cps) frequencies it is possible that “real” waveforms such as i.(f) above cannot be observed. However, at lower frequencies ( < 10’ cps) “real” waveforms may be observed on oscilloscopes. Thus one has the seemingly anomalous situation in which two waveforms appear different, but are coherent. We must therefore distinguish between coherence and similarity. It should be noted that a *:‘3 phase shift of all Fourier components converts nil) to s(f) and so the 1 For stationary fields the integral defining the conjugate field may diverge. To avoid this difficulty one may first assume that the field exists only for a finite time int,crval - 7’ 5 ts 7 and one can proceed to the limit I‘I--) m at the end of the calcnlat,ions.
74
STKE
I FE I<
definition (3) includes similarity if we permit devices which shift all relevant frequencies equally. It is indeed possible to construct (at least at, low enough frequencies) an electric filter which approximately shifts all frequencies in a finite band by a/2 (4). In the following we will continue to employ (3) as the definition of coherence. Eyuation (3) may he rewritten in the form, c.f. ref. 1,
(5) and by defining
one has the conditions (7) and (I’l(t III.
,2 completely
+ 7)15’(r))
= (I’dt
COMPLETEI,\i
+ T)Ir-(t))
COHEREST
= 0.
(8)
FIELI)S
coherent field for a region R is one in which 1 Ytj(Tij)J
= 1
for all pairs of points Pi and Pj in R for a least OIW value of 7ij . This is the definition given by 1Iandel and Wolf ( I ), but does not agree with Parrent’s (3). He requires / y12( T) ( = 1 for all T, which is perhaps too restrictive since it appears to admit only monochromatic fields. It is convenient to employ a reference point P, and write 7i0 = 7;) so that = Ti It is generally accepted (I, 2) that a completely coherent quasimonochromatic field permits the decomposition
Tij
Tj.
where
However, since I’iJ(T
71j
+ 2?TiL) =
yij(
Tij)
from (9) that one may select = I.
COMPLETELY
COHERENT
I~.-\I~I.~TION
Then Eq. (6) enables us to mrit,e l’,(f
+ ZkT;)
= I’,(f)
+ IV”(f),
where (IId t)rr;,*(t),
= 0,
and q’,(~f
It follows
+
7)11’“*(f))
=
0.
then, that Y,,(T)
= Y!,(T + ST/b)
and yI,( T) is periodic and monochromatic, i.e., it has a single d-function spectrum at 8. Thus the decomposition (9) holds only for strictly monochromatic radiation, but the mutual coherence function factors in the form ( 10) for all fully coherent radiation. This is because all such radiation exhibits cross-spectral purity ( 5), as is shown below. We write
and employ
( 6). One then finds
and r,j(T +
Tsj)
= It”It”ytj(
Tzj)yjj(
T).
(10)
The use of ( 9) has in fact assumed that y,,(T) = e-“+, In the foregoing, the function II’( 0, defined by (6), was employed. It would be very convenient to have It’(t) = 0, and indeed this will be assumed in the remainder of the work. To justify this assumption we note that W(t) is a realization of a process whose mean and mean square values are both zero, cf. EC{. (i). Thus, in the duration 7’, the lengt’h of time, ?IP , for which 1 Il’if)l
> E
is such that lim (7’P( e) /T) T-+% for any E > 0.
= 0,
Consider now a completely coherent field whose realization point is the analytic signal j(f). Since according to the above the disturbance at every other point must have the same time possible shift and complex proportionality factor, the most field may be expressed in the form G(.r, y, 2, f) = 11(x, y, z),f[f -
at some reference made assumption variation, with a general coherent
11(x, y, 2) “Cl
(.ll)
where
in a source free region. It has been assumed that all functions are differentiable as required and the index of refraction, n, depends only on position. The result of writing f( t - U/C) = j( [) and substituting (11 ) into ( 12) is
ffh Equation lut,ion
- ; f (I&
+ 2Vh.Vu) + $ Jf& (vu .VlL - ,?a’)= 0.
( 13) has a large class of solutions .f(<) = j(f)
= Ae’“’
including
the monochromatic
(13) so-
+ Be?,
which is obtained by setting u(x, y, Z) = 0. Of somewhat greater interest are those cases in which .f( <) is not determined by (13), thus permitting arbitrary source fluctuations. Then ( 18) must be satisfied identically by requiring that the coefficients of .f, r/,f!d<, and kff!dS,’ vanish individually, i.e., hiVu.Vu
- n”) = 0,
hV2u + 2VU’Vh
= 0,
( I-la) (l-lb)
and v’h = 0.
(1Jc)
It is interesting to note that these equations are identical with the equations of geometric optics. Physically this occurs because arbitrary source fluctuations permit discontinuities in the field, which propagate according to the laws of geometric optics (6). Friedlander (7) has considered systems of equations such as (Ma)-( 14~) with n constant, say n = I. He has shown that the most general solutions for the u = const. surfaces are the cyclides of Dupin (8), which include, the plane, sphere, cylinder, cone, torus, etc. 1Iost of these surfaces require that h(.c, y, z) be multivalued (7), but G(x, y, z, f) must be single valued. Thus for these to be
admissible solutions an obstacle must Icor example, nh~~ u = 1’ in cylindrical
be present to assure siriglc-valuedness. coordinates
and a screen must wstrict thr range of 0 t’o less In the cast of a finit’c volume of so~uws in solutions are allowed in the cstJerior region and cyclides, planes, and spheres. The first of these “c(2 + y? + 2”) + (2 + /j)( I -
than 2~. empty space only single valued onr is then limited to parabolic is defitwd hy ( 9)
IL) - z’( 1 + u) -
(.r - 1 -
(15) 11) (~1+ Il.)- = 0,
where 1 is an arbitrary cwlstjant. l“or large values of ZLtltrsc surfaws do not approach spheres or equivalcutly the Sommerfeld radiation condition is not satisfied, i.e., these solutions arc not permissible in our problem. l’lane waves are similarly eliminated, but we note in passing that the solution is then h(r,
y),f(i - z/c),
where C’h.(.r, 7J) = 0. Finally,
spherical waves are expressible
in the form (1Ga)
r
sin e fj
sin e $ (
>
”
+ i$ = 0. (0
(1Gh i
However, the only finite solut’ion of i IGh) at 0 = 0 is w = const., so that the only coherent field produced hy randomly fluctuating sources in a finite volume of empt#y space is the spherical wave
; 82 - I./C),
(17)
which includes the plane wave as a limiting case. If a coherent source is defined as one which produces a coherent field throughout the source region, then it is evident that the fields in the source free region need not, he fully coherent. Furthermore the decomposition of an incoherent source into infinitesimal sources producing coherent fields is equivalentj to stating that these sources produce waves of t’he form ( 17).
78
STREIFEK .kXNoWLEDGMENT
The author wishes to thank Drs. E. Wolf Iat)ing to various aspects of this work.
RECEIVED:
July
15,
and D. S. Ruchkiu
for several
discussions
re.
1963 REFERENCES
1, I,. M.INDEL .GD E. WOI~F, J. Opt. Sot. :lm. 51, 815 (1961). 2. (i. B. PARRENT, J. Opt. Sot. ilm. 49, 787 (1959). 3. G. B. PARRENT, Opt. .4cta 6, 285 (1959). 4. J. E. STORER. “Passive Network Synthesis,” p. 298. McGraw--Hill, Sew York, 1957. 5. L. MANUEL, J. Opt. Sot. Am. 51, 1342 (1961). 6. M. BORN AND E. WOLF, “Principles of Optics,” p. 108 aud Appendix \I. Pergaruon: London, 1959. i'. F. (i. FRIEDL.INDER, Proc. Vumb~itlge Phil. Sot. 43, 366 (1947). 8. A. R. FORSYTH “I)ifferentinl Geometry,” p. 324, Cambridge ITniv. Press, 1912; or A. C.\YLEY, “Maihematical Papers,” vol. IS, p. 64. Cambridge Ilniv. Press, 1896. 9. A. R. FORSYTH, lot. cit. p. 328.