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Generalized notion of the coherence area
Volume
5, number
OPTICS
4
GENERALIZED
July 1972
COMMUNICATIONS
NOTION
OF THE COHERENCE
AREA
A. ZARDECKI *, C. DELISLE and J. BURES Laboratoire de Recherches
en Optique et Laser, Dkpartement Qu&ec IO, Canada
de Physique,
Universite'Laval,
Received 20 March 1972 Revised manuscript received 21 April 1972
The notion of a coherence area is introduced on the basis of photoelectron earlier results, the case of a source of arbitrary shape and intensity distribution made between the number of degrees of freedom and the number of coherent
distribution analysis. In contrast is considered. A clear distinction elements.
with is
It has been shown previously [ 1,2] that for some simple geometrical configurations, under the assumptions that the light is quasi-monochromatic and the intensity distribution of the incoherent source is uniform, the area of coherence is given as SC = X;D2/Ss
(1)
,
where X0 is the mean wavelength, S, is the area of the source, and D is the distance from the source to the observation plane. This result was obtained earlier in another context by many authors [3-61. In connection with this problem see also Mandel and Wolf [7]. It is worth noting that a similar expression appears in the calculation of the number of resolvable area elements of the source [8]. The purpose of this paper is to show that formula (1) is valid in the case of any incoherent quasi-monochromatic plane source. Then, we shall generalize that formula for the case where the intensity distribution over the surface of the source is non-uniform. We will base our derivation on considering the photoelectron counting distribution in the plane of an extended photodetector illuminated by the light coming from an incoherent quasi-monochromatic thermal source [2]. When the observation time T is much shorter than the coherence time of light, the number n of photoelectrons is described, correctly up to the second moment, by the probability distribution p(n,T)=
{l-(n+N)/n!r(N)}
[l +(H>/N]-~
[l +N/(n)]-”
,
(2)
where the parameter N can be identified with the number of statistical degrees of freedom of the light. Explicitly N is given as an integral over the surface of the detector, S,j, N-’
= (l/S;)
j- j- Icl(r1,r2)12 d2r1 d2r2 ,
(3)
sd sd
where p(rI ,r2) is the complex degree of coherence and it is assumed that the distance between the source and the detector is so large that we can regard the detector illumination, in the region where /.I@~,rz) is appreciably different from zero, as being effectively uniform. In this approximation the degree of coherence ~(rl ,r2) is given as a normalized Fourier transform of the inten* On leave of absence
from Institute
of Physics,
The Warsaw Technical
University,
Warsaw,
Poland.
Volume 5, number 4
OPTICS COMMUNICATIONS
July 1972
sity function Z([,n) of the source [9]
s, (4)
where k, = 2n/ho, D is the distance between the source and the detector, and rI = (xl,yl), f2 = (x2,y2). The phase factor \kI 2 is of no interest in our problem. Eq. (2) is formally identical with the equation dreived by Mandel [lo] when light is detected at a single point of space. In that case the parameter N is then defined differently from (3). The problem of statistical degrees of freedom was dealt with earlier by Gabor [ 111. If the detector is very small, the exponential expression in the integrand of (4) can be set equal to unity, and it follows from (3) that N= 1. In this case the distribution (2) becomes the Bose-Einstein distribution. If, on the other hand, we let formally the detector surface tend to infinity, N becomes independent of the detector shape. The passage to the limit may be justified by noting that the degree of coherence vanishes effectively outside a small region in the vicinity of the origin of the (XJ) plane. For the purpose of computation we choose a detector of square shape with sides parallel to the x and y axes of the coordinate system whose origin coincides with the center of the square. After substituting (4) into (3) we perform first the integration over the variablesxl, yl and
Fig. 1. Plot of both the number of degrees of freedom, N, and the number of coherent elements, Nc, as function of the detector surface Sd, for a circular geometry of the source and detector. The intensity distribution of the source is uniform. The detector surface is expressed in units of $D2/Ss. The straight line which corresponds to NC, is an asymptote to the plot of N. The area of coherence, SC, is given in these units as Sd = 1.
Volume 5, number 4
OPTICS COMMUNICATIONS
x2, y2. In the large detector limit this yields a factor involving a delta function permitting tor to a single integral over the surface of the source. The final result of these calculations
N-’
= &D2/S,)
Jr2(Lrl)dl
5
July 1972
to reduce the numerais
dd[~WWh12 5
(5)
and it shows that N is proportional to the surface of detection, the latter being assumed to be very large. On the basis of this proportionality, we define the area of coherence as the area of detection per statistical degree of freedom, provided the observation time is much smaller than the coherence time of light. It follows from (5) that S c = h2D2 J12(&?) 0
%
dg dg/[
~~(WW~l* %
(6)
In the special case when the intensity over the surface of the source is uniform, eq. (1) is recovered. Fig. 1 shows a typical plot of eq. (3) corresponding to circular geometrical arrangement of the source and detector. It is seen that each value of the detector area defines a certain number of degrees of freedom N. According to the physical interpretation N cannot be smaller than unity. The asymptote to the curve of N versus S, determines, what may be called the number of the coherent elements of the detector, N,. It is to be noted that the value of the parameter N obtained experimentally, for example from the measurements of the second moment of p(n, 7’), does not correspond to the number of elementary coherent elements making up the detector but to the number of degrees of freedom of the light. In fact, both N and N, coincide only in the limit of an infinitely large detector. For small values of the detection area N, tends to zero. The extension of the obtained results for the case of an arbitrary observation time will be a subject of a separate study. The support of National Research Council of Canada is gratefully
acknowledged.
References 111 C. Delisle, J. Bures and A. Zardecki, J. Opt. Sot. Am. 61 (1971) 1589. [2] J. Bures, C. Delisle and A. Zardecki, Can. J. Phys. 50 (1972) 760. 131 A.T. Forrester, R.A. Gudmundsen and P.O. Johnson, Phys. Rev. 99 (1955) 1691. [4] R. Hanbury Brown and R.Q. Twiss, Proc. Roy. Sot. 243A (1958) 291. [ 51 A. Kastler,in: Quantum electronics, Proc. Third International Congress, eds. N. Bloembergen and P. Grivet (Dunod, Paris. 1964) p. 3. 161H.Z. Cummins and H.L. Swinney, Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 135. [71 L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. [81 H.A. Haus, International School of Physics. “Enrico Fermi” XL11 Course, ed. R.J. Glauber (Academic Press, New York, 1969) p. 111. [91 M. Born and E. Wolf, Principles of optics (Pergamon Press, London, 1970) p. 510. IlO1 L. Mandel, Proc. Phys. Sot. (London) 74 (1959) 233. illI D. Gabor, Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 1961) p. 107.
5, number
OPTICS
4
GENERALIZED
July 1972
COMMUNICATIONS
NOTION
OF THE COHERENCE
AREA
A. ZARDECKI *, C. DELISLE and J. BURES Laboratoire de Recherches
en Optique et Laser, Dkpartement Qu&ec IO, Canada
de Physique,
Universite'Laval,
Received 20 March 1972 Revised manuscript received 21 April 1972
The notion of a coherence area is introduced on the basis of photoelectron earlier results, the case of a source of arbitrary shape and intensity distribution made between the number of degrees of freedom and the number of coherent
distribution analysis. In contrast is considered. A clear distinction elements.
with is
It has been shown previously [ 1,2] that for some simple geometrical configurations, under the assumptions that the light is quasi-monochromatic and the intensity distribution of the incoherent source is uniform, the area of coherence is given as SC = X;D2/Ss
(1)
,
where X0 is the mean wavelength, S, is the area of the source, and D is the distance from the source to the observation plane. This result was obtained earlier in another context by many authors [3-61. In connection with this problem see also Mandel and Wolf [7]. It is worth noting that a similar expression appears in the calculation of the number of resolvable area elements of the source [8]. The purpose of this paper is to show that formula (1) is valid in the case of any incoherent quasi-monochromatic plane source. Then, we shall generalize that formula for the case where the intensity distribution over the surface of the source is non-uniform. We will base our derivation on considering the photoelectron counting distribution in the plane of an extended photodetector illuminated by the light coming from an incoherent quasi-monochromatic thermal source [2]. When the observation time T is much shorter than the coherence time of light, the number n of photoelectrons is described, correctly up to the second moment, by the probability distribution p(n,T)=
{l-(n+N)/n!r(N)}
[l +(H>/N]-~
[l +N/(n)]-”
,
(2)
where the parameter N can be identified with the number of statistical degrees of freedom of the light. Explicitly N is given as an integral over the surface of the detector, S,j, N-’
= (l/S;)
j- j- Icl(r1,r2)12 d2r1 d2r2 ,
(3)
sd sd
where p(rI ,r2) is the complex degree of coherence and it is assumed that the distance between the source and the detector is so large that we can regard the detector illumination, in the region where /.I@~,rz) is appreciably different from zero, as being effectively uniform. In this approximation the degree of coherence ~(rl ,r2) is given as a normalized Fourier transform of the inten* On leave of absence
from Institute
of Physics,
The Warsaw Technical
University,
Warsaw,
Poland.
Volume 5, number 4
OPTICS COMMUNICATIONS
July 1972
sity function Z([,n) of the source [9]
s, (4)
where k, = 2n/ho, D is the distance between the source and the detector, and rI = (xl,yl), f2 = (x2,y2). The phase factor \kI 2 is of no interest in our problem. Eq. (2) is formally identical with the equation dreived by Mandel [lo] when light is detected at a single point of space. In that case the parameter N is then defined differently from (3). The problem of statistical degrees of freedom was dealt with earlier by Gabor [ 111. If the detector is very small, the exponential expression in the integrand of (4) can be set equal to unity, and it follows from (3) that N= 1. In this case the distribution (2) becomes the Bose-Einstein distribution. If, on the other hand, we let formally the detector surface tend to infinity, N becomes independent of the detector shape. The passage to the limit may be justified by noting that the degree of coherence vanishes effectively outside a small region in the vicinity of the origin of the (XJ) plane. For the purpose of computation we choose a detector of square shape with sides parallel to the x and y axes of the coordinate system whose origin coincides with the center of the square. After substituting (4) into (3) we perform first the integration over the variablesxl, yl and
Fig. 1. Plot of both the number of degrees of freedom, N, and the number of coherent elements, Nc, as function of the detector surface Sd, for a circular geometry of the source and detector. The intensity distribution of the source is uniform. The detector surface is expressed in units of $D2/Ss. The straight line which corresponds to NC, is an asymptote to the plot of N. The area of coherence, SC, is given in these units as Sd = 1.
Volume 5, number 4
OPTICS COMMUNICATIONS
x2, y2. In the large detector limit this yields a factor involving a delta function permitting tor to a single integral over the surface of the source. The final result of these calculations
N-’
= &D2/S,)
Jr2(Lrl)dl
5
July 1972
to reduce the numerais
dd[~WWh12 5
(5)
and it shows that N is proportional to the surface of detection, the latter being assumed to be very large. On the basis of this proportionality, we define the area of coherence as the area of detection per statistical degree of freedom, provided the observation time is much smaller than the coherence time of light. It follows from (5) that S c = h2D2 J12(&?) 0
%
dg dg/[
~~(WW~l* %
(6)
In the special case when the intensity over the surface of the source is uniform, eq. (1) is recovered. Fig. 1 shows a typical plot of eq. (3) corresponding to circular geometrical arrangement of the source and detector. It is seen that each value of the detector area defines a certain number of degrees of freedom N. According to the physical interpretation N cannot be smaller than unity. The asymptote to the curve of N versus S, determines, what may be called the number of the coherent elements of the detector, N,. It is to be noted that the value of the parameter N obtained experimentally, for example from the measurements of the second moment of p(n, 7’), does not correspond to the number of elementary coherent elements making up the detector but to the number of degrees of freedom of the light. In fact, both N and N, coincide only in the limit of an infinitely large detector. For small values of the detection area N, tends to zero. The extension of the obtained results for the case of an arbitrary observation time will be a subject of a separate study. The support of National Research Council of Canada is gratefully
acknowledged.
References 111 C. Delisle, J. Bures and A. Zardecki, J. Opt. Sot. Am. 61 (1971) 1589. [2] J. Bures, C. Delisle and A. Zardecki, Can. J. Phys. 50 (1972) 760. 131 A.T. Forrester, R.A. Gudmundsen and P.O. Johnson, Phys. Rev. 99 (1955) 1691. [4] R. Hanbury Brown and R.Q. Twiss, Proc. Roy. Sot. 243A (1958) 291. [ 51 A. Kastler,in: Quantum electronics, Proc. Third International Congress, eds. N. Bloembergen and P. Grivet (Dunod, Paris. 1964) p. 3. 161H.Z. Cummins and H.L. Swinney, Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 135. [71 L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. [81 H.A. Haus, International School of Physics. “Enrico Fermi” XL11 Course, ed. R.J. Glauber (Academic Press, New York, 1969) p. 111. [91 M. Born and E. Wolf, Principles of optics (Pergamon Press, London, 1970) p. 510. IlO1 L. Mandel, Proc. Phys. Sot. (London) 74 (1959) 233. illI D. Gabor, Progress in optics, Vol. 1, ed. E. Wolf (North-Holland, Amsterdam, 1961) p. 107.