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Photocount statistics of Gaussian light of arbitrary spectral profile
Volume 35A, number 2
PHYSICS LETTERS
solutions, and that the deviations are due to short-range ordering or clustering. It is further suggested that clustering will decrease the resistivity and thus result in values of A greater than 1; short-range ordering may however either increase or decrease the resistivity and may therefore produce values of A either greater or less than 1. For all the systems shown in figure 1, A is greater than unity and thus may be due to either clustering or short-range ordering. Effects of this type are known to be present in some systems: Stringer [4] has reviewed the available information for the systems Cu-Zn and Cu-Ge, and has remarked on the probable importance of local ordering effects on the galvanomagnetic properties of alloys: the correspondence shown in fig. 1 would seem to lend strong support to this view. However, it must be pointed out that
PHOTOCOUNT
STATISTICS
ARBITRARY
17 May 1971
the available data for silver- and gold-based solid solutions do not show a similar relationship. Thanks are due to Professor J. Stringer for interesting discussions on this subject and to the Science Research Council for the provision of a Research Studentship.
References [1] J. 0. Linde, Helv. Phys. Acta 41 (1968) 1007. [2] A.H: Hitchcock and J. Stringer, Phys. Rev., to be [3] W KOster and H.-P. Rave, Z. Metallk. 55 (1964) 750. [41J. Stringer, in Phase stability of metals and alloys, eds. P. S. Rudman. J. Stringer and R. I. Jafee (McGraw-Hill, New York. 1967) pp 165-193.
OF GAUSSIAN LIGHT OF
SPECTRAL
PROFILE
S.K.SRINWASAN and S. SUKAVANAM Department of Mathematics, Indian Institute of Technology, Madras—36. india Received 18 March 1971 The letter presents a general method of arriving at the generating function of the photocount distribution corresponding to Gaussian light beams of arbitrary spectral profile and centre frequency.
In this letter we wish to present a general method of arriving at the generating function of
the photocount distribution, thus providing a generalization of the results obtained by Jakeman and Pike [1]. We start with the formula for the generating function Q(s) given by [1] Q(s)
=
(exp(-Es))
=
11(1
+5
Ak)
(1)
where (E)/T is the mean intensity, T the sampling time and the Ak’s are the eigenvalues of the equation
o
=
=
A~(t) .
~P(P)=
f
t(t)exp(-p/)d/,
1exp{(z-P)T}ip(z)-t~i(p)
~‘.
z -p
“
cr—ion
(2)
Eq. (2) can be solved more generally by replacing exp (- r I - t’ ) by a real valued function f defined on the positive real axis so that it vanishes for
ff(t)exp(-pt)dt,
the genetalization of eq. (2) leads to cr+ 1 ~ ~.*, ~ 2iri
T
f exp (-nt —t’ I) ~(t’)dt’
large values of its argument. This would correspond either to a super-position of a number of Lorentzian profiles of arbitrary widths or to a single beam of arbitrary profile. Defining f*(p) and ~‘(p)by T T (3)
(4)
~p(—z)—~p(p) +
~
}dz=x~p(p)
We note that) * (z) is analytic in the half -plane Ri z > 0. For purposes of computation we shall 81
Vu~ume 35A. number
I’I-IYSICS
2
assume 1*(z) = g(z)r 1i(z) where h(z) is a p~i~nomial of degree n and g(z) of degree less than e Observing that i~i(Z) is an entire function and making use of its asymptotic behaviour we evaluate the line integral for fixed P (as in [2]) by choosing u to be greater than max(0, Ri p. Ri(-p)) and finally obtain Ji(p) Iz(-p) ~
(akexp(-pT)
~
k=l ~/i(p)
h(-p) -g(p) h(-p)-g(-p)/i(p)
!7Mav 1971
LETT
(5)
0k and hk are respectively the (ik - l)th where derivatives of
ure arising from the dependence of pt’s the zeros of the denominator of the right hand side of eq. (5) on ~. However the determinant is an alternant in P 1, P2,. P2,r Thus we can construct an entire function P(~)by multiplying or dividing F(~) by a known suitable alternant. In this case it is easy to divide F(~) by a simple alternant I) whose elements are given by D11 = - ~ Using the properties of g(p) and /i(p)it can be shown that for largevaluesof~. P1isoftheorderof 1/2, p~being any one of the zeros of the denominator of the right hand side of eq. (5). Thus theHadamard’s order of thefactorization entire function P(~) is(eq. half. of theorem 3) Use yields
[1/(l 1,- 1) ] (z - 2k)
exp (2T)~(4t*(z)/(z ~p)
P(~)
and [1 (li-I)
1 (~~)
k~2)1
2k and 21,
(2)/(r
P(0)
H (1-
~
where the ~~‘s are the zeros of P(~). This combined with eq. (1) yields G(s) = P(0)/P(-,sE/T) (7)
4~)
Z
evaluated at 2 = 2, . ‘ are the distinct zeros of Iz(z) with multiplicities l1, l~,. , . l~ respectively. Thus m~i(p)is determined for any given p provided of the derivatived of 2k the andvalues -2k are determined, ~ atWethe next points note that the denominator of the right hand side of eq. (5) is a polynomial in p of degree 2n and hence has 2n zeros. However qi~) is an entire function and as such the numerator should vanish at these points. These conditions yield a set of 2n homogeneous linear equations for the values of all the derivatives of ~i at 2k and -zk occurring in °k and Ilk. In order that this solution be non-trivial we demand that the determinant of the coefficients should be zero. Denoting the determinant of the coefficients by F(1/X) we note that F(l/X) 0 gives the eigenvalues of the basic integral equation (2). Putting ~ we observe that F(~) is not an entire function of ~ . This is due to the multiple-valued nat —
We wish to observe that eq. (2) can be solved even more generally by replacing f by a complex valued function to accommodate the physically interesting corresponding mixtures of beams of case arbitrary profile andtocentre frequencies. We have cerified that this technique yields the solution corresponding to a mixture of coherent and incoherent beams discussed earlier in ref. [1]. The method is also capable of yielding the generating function of the photocounts of light beams arbitrarily polarized. Details of these caluclations will be published elsewhere.
References
1’2J
E.Jakeman and 6.11. Pike, .J. Phys.A2 ~1969) 115. S. K. Srinivasan and 11. Subramanjan. J. Math. Phvs. Sci. ~ i1969) 221. [3J 6. hub, Analytic function theory. Vol. 2 iBlaisdell
l’ublishin 6 Company, 1962).
00*0
82
(6)
*
PHYSICS LETTERS
solutions, and that the deviations are due to short-range ordering or clustering. It is further suggested that clustering will decrease the resistivity and thus result in values of A greater than 1; short-range ordering may however either increase or decrease the resistivity and may therefore produce values of A either greater or less than 1. For all the systems shown in figure 1, A is greater than unity and thus may be due to either clustering or short-range ordering. Effects of this type are known to be present in some systems: Stringer [4] has reviewed the available information for the systems Cu-Zn and Cu-Ge, and has remarked on the probable importance of local ordering effects on the galvanomagnetic properties of alloys: the correspondence shown in fig. 1 would seem to lend strong support to this view. However, it must be pointed out that
PHOTOCOUNT
STATISTICS
ARBITRARY
17 May 1971
the available data for silver- and gold-based solid solutions do not show a similar relationship. Thanks are due to Professor J. Stringer for interesting discussions on this subject and to the Science Research Council for the provision of a Research Studentship.
References [1] J. 0. Linde, Helv. Phys. Acta 41 (1968) 1007. [2] A.H: Hitchcock and J. Stringer, Phys. Rev., to be [3] W KOster and H.-P. Rave, Z. Metallk. 55 (1964) 750. [41J. Stringer, in Phase stability of metals and alloys, eds. P. S. Rudman. J. Stringer and R. I. Jafee (McGraw-Hill, New York. 1967) pp 165-193.
OF GAUSSIAN LIGHT OF
SPECTRAL
PROFILE
S.K.SRINWASAN and S. SUKAVANAM Department of Mathematics, Indian Institute of Technology, Madras—36. india Received 18 March 1971 The letter presents a general method of arriving at the generating function of the photocount distribution corresponding to Gaussian light beams of arbitrary spectral profile and centre frequency.
In this letter we wish to present a general method of arriving at the generating function of
the photocount distribution, thus providing a generalization of the results obtained by Jakeman and Pike [1]. We start with the formula for the generating function Q(s) given by [1] Q(s)
=
(exp(-Es))
=
11(1
+5
Ak)
(1)
where (E)/T is the mean intensity, T the sampling time and the Ak’s are the eigenvalues of the equation
o
=
=
A~(t) .
~P(P)=
f
t(t)exp(-p/)d/,
1exp{(z-P)T}ip(z)-t~i(p)
~‘.
z -p
“
cr—ion
(2)
Eq. (2) can be solved more generally by replacing exp (- r I - t’ ) by a real valued function f defined on the positive real axis so that it vanishes for
ff(t)exp(-pt)dt,
the genetalization of eq. (2) leads to cr+ 1 ~ ~.*, ~ 2iri
T
f exp (-nt —t’ I) ~(t’)dt’
large values of its argument. This would correspond either to a super-position of a number of Lorentzian profiles of arbitrary widths or to a single beam of arbitrary profile. Defining f*(p) and ~‘(p)by T T (3)
(4)
~p(—z)—~p(p) +
~
}dz=x~p(p)
We note that) * (z) is analytic in the half -plane Ri z > 0. For purposes of computation we shall 81
Vu~ume 35A. number
I’I-IYSICS
2
assume 1*(z) = g(z)r 1i(z) where h(z) is a p~i~nomial of degree n and g(z) of degree less than e Observing that i~i(Z) is an entire function and making use of its asymptotic behaviour we evaluate the line integral for fixed P (as in [2]) by choosing u to be greater than max(0, Ri p. Ri(-p)) and finally obtain Ji(p) Iz(-p) ~
(akexp(-pT)
~
k=l ~/i(p)
h(-p) -g(p) h(-p)-g(-p)/i(p)
!7Mav 1971
LETT
(5)
0k and hk are respectively the (ik - l)th where derivatives of
ure arising from the dependence of pt’s the zeros of the denominator of the right hand side of eq. (5) on ~. However the determinant is an alternant in P 1, P2,. P2,r Thus we can construct an entire function P(~)by multiplying or dividing F(~) by a known suitable alternant. In this case it is easy to divide F(~) by a simple alternant I) whose elements are given by D11 = - ~ Using the properties of g(p) and /i(p)it can be shown that for largevaluesof~. P1isoftheorderof 1/2, p~being any one of the zeros of the denominator of the right hand side of eq. (5). Thus theHadamard’s order of thefactorization entire function P(~) is(eq. half. of theorem 3) Use yields
[1/(l 1,- 1) ] (z - 2k)
exp (2T)~(4t*(z)/(z ~p)
P(~)
and [1 (li-I)
1 (~~)
k~2)1
2k and 21,
(2)/(r
P(0)
H (1-
~
where the ~~‘s are the zeros of P(~). This combined with eq. (1) yields G(s) = P(0)/P(-,sE/T) (7)
4~)
Z
evaluated at 2 = 2, . ‘ are the distinct zeros of Iz(z) with multiplicities l1, l~,. , . l~ respectively. Thus m~i(p)is determined for any given p provided of the derivatived of 2k the andvalues -2k are determined, ~ atWethe next points note that the denominator of the right hand side of eq. (5) is a polynomial in p of degree 2n and hence has 2n zeros. However qi~) is an entire function and as such the numerator should vanish at these points. These conditions yield a set of 2n homogeneous linear equations for the values of all the derivatives of ~i at 2k and -zk occurring in °k and Ilk. In order that this solution be non-trivial we demand that the determinant of the coefficients should be zero. Denoting the determinant of the coefficients by F(1/X) we note that F(l/X) 0 gives the eigenvalues of the basic integral equation (2). Putting ~ we observe that F(~) is not an entire function of ~ . This is due to the multiple-valued nat —
We wish to observe that eq. (2) can be solved even more generally by replacing f by a complex valued function to accommodate the physically interesting corresponding mixtures of beams of case arbitrary profile andtocentre frequencies. We have cerified that this technique yields the solution corresponding to a mixture of coherent and incoherent beams discussed earlier in ref. [1]. The method is also capable of yielding the generating function of the photocounts of light beams arbitrarily polarized. Details of these caluclations will be published elsewhere.
References
1’2J
E.Jakeman and 6.11. Pike, .J. Phys.A2 ~1969) 115. S. K. Srinivasan and 11. Subramanjan. J. Math. Phvs. Sci. ~ i1969) 221. [3J 6. hub, Analytic function theory. Vol. 2 iBlaisdell
l’ublishin 6 Company, 1962).
00*0
82
(6)
*