Modal expansion of thin annular sources with Schell-model angular correlation function
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J. Opt. 14 (2012) 035701 (6pp) doi:10.1088/2040-8978/14/3/035701
Modal expansion of thin annular sources with Schell-model angular correlation
function
Massimo Santarsiero
1, Franco Gori
1, Riccardo Borghi
2and Giorgio Guattari
21Dipartimento di Fisica, Universit`a Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy
2Dipartimento di Elettronica Applicata, Universit`a Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy
E-mail:[email protected]
Received 24 November 2011, accepted for publication 3 January 2012 Published 27 January 2012
Online atstacks.iop.org/JOpt/14/035701 Abstract
A general procedure is presented for the evaluation of the modes of a thin annular scalar source, whose angular mutual intensity is of the Schell-model type. Starting from the knowledge of the modes, the coherence properties of the field propagated from the source in paraxial conditions can be evaluated. When the propagated field is collimated by a suitable converging lens, presented results apply to the synthesis of propagation-invariant partially coherent beams.
Keywords: partially coherent sources, modal theory of coherence, diffraction-free beams
1. Introduction
According to the modal theory of coherence [1], any partially coherent source can be thought of as the superposition of a (possibly infinite) number of mutually uncorrelated and perfectly coherent sources. The fields associated with such sources and their relative power weights can be evaluated in a very general way by calculating eigenfunctions and eigenvalues of a suitable Fredholm integral equation.
Since its introduction in scalar coherence theory, modal expansion has represented a powerful tool in the characterization of partially coherent sources, as well as in the study of the fields they radiate. On the one hand, in fact, the distribution of the eigenvalues gives information about global coherence features of the source [2, 3]; on the other hand, the knowledge of the coherence features of the field propagated from a partially coherent source is accomplished by evaluating the propagation of a number of perfectly coherent fields [4–9], which is significantly easier than dealing with partially coherent fields. As a counterpart of their undoubted usefulness, finding the modes of a partially coherent source in an analytical way is not in general a simple task and, even in the case of scalar sources, closed-form
expressions have been obtained only in a relatively small number of cases [4,6,10–15].
An interesting class of partially coherent planar sources is that of the annular sources, whose field is confined within a thin annular region. The interest for such sources mainly stems from their use in the synthesis of fields endowed with peculiar propagation properties. In the limit of perfect spatial coherence, in fact, annular sources have been used to synthesize the so-called diffraction-free, or Bessel beams [16,17], which do not change their transverse shape upon propagation. To obtain these beams, the field radiated from the source has to be collimated by a converging lens of suitable focal length. When the annular source is not perfectly coherent from the spatial point of view, the propagated beam is partially coherent but its propagation-invariance properties are preserved, in the sense that not only the irradiance profile, but also the correlations between two fixed points are exactly the same in every transverse plane [18]. Due to their potential use in practical applications, partially coherent diffraction-free beams have been deeply studied in recent years [19–26].
In the case of a partially coherent annular source, the modal structure has been found only in the limit of
2040-8978/12/035701+06$33.00 1 c 2012 IOP Publishing Ltd Printed in the UK & the USA
J. Opt. 14 (2012) 035701 M Santarsiero et al
complete incoherence [10], or when the source is angularly homogeneous, i.e. when the correlation function between the fields at two points of the annulus depends only on the difference of their angular coordinates [15]. In both cases the irradiance along the annulus is supposed to be uniform and the modes of fields propagated from the source turn out to be higher-order Bessel beams. In the present paper, we extend the previous analyses to thin annular sources with a Schell-model angular correlation function, i.e. sources with an angularly shift-invariant degree of coherence, but with a generally position-dependent irradiance. In particular, we give a general procedure for evaluating the modes of the source as well as those of the propagated field and present an example where all calculations can be performed in a closed-form simple analytical way.
2. Preliminaries
The coherence properties between two points specified by the vectors ⇢1and ⇢2across a planar quasi-monochromatic source can be described by the mutual intensity function, namely J(⇢1,⇢2,0), defined as [1]
J(⇢1,⇢2,0) = hV(⇢1,0, t)V⇤(⇢2,0, t)i, (1) where the plane z = 0 has been chosen as the source plane and V(⇢, z, t) is the analytic signal associated with the field at the point (⇢, z) and time t. By assuming the process to be stationary and ergodic, the angle brackets can be thought of as denoting a time average. Alternatively, the cross-spectral density [1] could be used but, for quasi-monochromatic light, the two approaches are equivalent.
In order for the function J to be representative of a physical light source, it has to be a non-negative definite kernel [1]. This means that the following condition
Z Z
J(⇢1,⇢2,0)f⇤(⇢1)f (⇢2)d2⇢1d2⇢2 0, (2) must hold for an arbitrary choice of the (well behaving) function f (⇢).
We further recall that by modal expansion we essentially mean the Mercer’s series [1]
J(⇢1,⇢2,0) =X
n
3n9n(⇢1,0)9n⇤(⇢2,0), (3) where 3n and 9n denote eigenvalues and eigenfunctions, respectively, of the homogeneous Fredholm integral equation
Z
J(⇢1,⇢2,0)9(⇢2,0) d2⇢2= 39(⇢1,0), (4) the integral being extended to the source plane. Because of the condition in equation (2), all the eigenvalues are non-negative.
According to equation (3), any partially coherent source can be thought of as arising from the superposition of a set of mutually uncorrelated, perfectly coherent, orthonormal fields, 9n(⇢, 0), each of them carrying the power 3n.
Here we deal with partially coherent scalar sources having the shape of an infinitely thin annulus of radius a. Their mutual intensity can be then written in the form
J(⇢1,⇢2,0) = K (⇢1 a) (⇢2 a)Ja('1, '2), (5)
where ⇢j= (⇢j, 'j), (j = 1, 2), are polar coordinates across the source plane, K is a constant factor having dimensions of an area [15], and is the Dirac function. The function Ja
gives account of the correlation properties along the annulus and will be referred to as the angular mutual intensity of the source. In fact, it must fulfil all the properties of a mutual intensity. In particular, because of the basic relation
J(⇢2,⇢1,0) = J⇤(⇢1,⇢2,0), (6) directly derived from the definition in equation (1), we see that Ja has to be Hermitian, that is Ja('2, '1)= J⇤a('1, '2).
Furthermore, the non-negativity property has to hold for Ja, i.e.
Z 2⇡
0
Z 2⇡
0 Ja('1, '2)f⇤('1)f ('2)d'1d'1 0, (7) for any choice of the function f ('), and a Mercer expansion can be written as
Ja('1, '2)=X
n
n8n('1)8⇤n('2), (8) with 8n(') and n eigenfunctions and eigenvalues, respec- tively, of the integral equation
Z 2⇡
0 Ja('1, '2)8('2)d'2= 8('1). (9) 3. Modal analysis
We limit ourselves to the cases in which Jatakes the following functional form:
Ja('1, '2)= s('1)s⇤('2)g('1 '2). (10) Following the terminology used in the case of general partially coherent sources [1], sources of this type will be denoted as Schell-model annular sources.
The optical intensity along the annulus turns out to be proportional to |s(')|2, so that we shall loosely refer to the quantity
I(') = |s(')|2 (11)
as the intensity of the source along the annulus. On the other hand, the function g represents the (shift-invariant) degree of coherence. We further suppose that g is regular enough to admit the Fourier expansion
g('1 '2)= X1
n= 1
nein('1 '2). (12) Sources of this type, but with I(') = constant, were studied in [15].
Because of the Hermiticity of Ja, the Fourier coefficients
n must be real. Furthermore, from the non-negativeness condition on Ja, we know that such coefficients cannot be negative [15].
On using the Fourier expansion in equation (12) in the definition in equation (10), the following expression is derived:
2
Ja('1, '2)= X1
n= 1
nein('1 '2)s('1)s⇤('2)
= X1
n= 1
n0Un('1)U⇤n('2), (13)
with
Un(')= s(')
pPtotein'; n0 = Ptot n, (14) where the fields Un(')have been normalized, the factor Ptot being proportional to the total power carried by the source, i.e.
Ptot= Z 2⇡
0 I(')d'. (15)
Although the sum in equation (13) may look like a Mercer expansion, this is not the case because the functions Un(')are normalized but they are not, in general, mutually orthogonal.
A significant exception is when the intensity is uniform along the annulus [15]. In such a case, in fact, we have
Z 2⇡
0 Un(')U⇤m(')d' = 1 2⇡
Z 2⇡
0 ei(n m)'d' = nm, (16) with nm being the Kronecker symbol. Then, the functions Unturn out to be the modes of the source, with eigenvalues coinciding with the coefficients n0.
Now we come back to the more general case, in which the intensity is not constant along the annulus. Even in this case the expansion in equation (13) can be read as follows. The annular source can be obtained on superposing a (possibly infinite) number of mutually uncorrelated annular coherent fields, whose angular part is denoted by Un('), with powers given by the set of coefficients n0. In general, such fields are not the modes of the source because they are not mutually orthogonal [27].
It is possible, however, to evaluate the modes starting from the expressions of the fields Un(')and the values of the
n0coefficients. To this aim, it is convenient to define a new set of fields, namely,
Vn(')=p 0
nUn('), (17)
so that the angular mutual intensity of the source takes the form
Ja('1, '2)= X1
n= 1
Vn('1)Vn⇤('2). (18) We recall that the Fourier coefficients n are non- negative, so that the n0 coefficients (see equation (14)) and their square roots are always real and non-negative.
Since the functions Vn vanish if the corresponding n coefficients vanish (see equation (17)), we can limit the series in equation (18) to those values of n that belong to the set of integers for which the value of nis different from zero. If N is the number of such non-vanishing coefficients, we denote them by nj(j = 1, . . . , N) and write
Ja('1, '2)= XN
j=1
Vnj('1)Vn⇤j('2). (19)
In practical applications, only the terms for which nis significantly different from zero can be considered.
Now, we expand a typical mode of the source as a linear combination of the Vnfunctions, i.e.
8(')= XN
j=1
cjVnj('), (20)
where cj are unknown coefficients, and use the definition of mode in equation (9). In such a way we obtain
XN j,`=1
Vn`('1)A`jcj= XN j=1
cjVnj('1), (21)
where the following matrix element has been introduced:
A`j= Z 2⇡
0 Vn⇤`(')Vnj(')d'
=p
n` nj
Z 2⇡
0 I(')e i(n` nj)'d', (22)
with `, j = 1, . . . , N, and the definitions in equations (14), (15) and (17) have been used. Note that in general the N ⇥ N matrix ˆA = {A`j} is Hermitian and, if the functions Vn are mutually orthogonal, it turns out to be diagonal. Furthermore, it is worth noting that the phase of the function s(') does not affect the evaluation of the A`jcoefficients.
If we now multiply both sides of equation (21) by Vn⇤m('1), with m = 1, . . . , N, and integrate them with respect to '1on the interval [0, 2⇡], we obtain
XN
`,j=1
Am`A`jcj= XN
j=1
Amjcj. (23)
On introducing the column vectors c = {cj}, the latter equation can be written in a more compact form as
ˆA( ˆAc) = ˆA( c), (24)
which is satisfied if
ˆAc = c. (25)
The latter is nothing but the eigenvalue equation for ˆA.
Therefore, the N eigenvalues of the angular mutual intensity are given by those of ˆA, while the corresponding modes can be obtained from the eigenvectors c(i) by means of equation (20). Up to normalization factors, the latter equation yields, together with equations (14) and (17),
8i(')= s(') XN
j=1
c(i)j p
njeinj', (26) which gives the expression of the modes of a general annular Schell-model source.
As a final remark, we note that the evaluation of the elements of the matrix ˆA is a rather easy task, because they coincide, up to a proportionality factor, with the Fourier coefficients of the function I('). The latter function, in fact, has to be periodic with period 2⇡ and, assuming its behavior
J. Opt. 14 (2012) 035701 M Santarsiero et al
as regular enough, can be written in the form I(') = X1
n= 1
nein', (27)
with
n= 1 2⇡
Z 2⇡
0 I(')e in'd'. (28)
On comparing equations (22) and (28) it is then readily seen that the elements of ˆA are
A`j= 2⇡p
n` nj n` nj. (29) Since the function I(') is real and positive, it must hold that n= n⇤and 0>0.
4. Propagation of the modes
The knowledge of the modes of a source allows one to evaluate in a rather simple way the coherence properties of the propagated field. In fact, there is no need to evaluate the expression of the propagated mutual intensity, which involves the solution of a four-variable integral [1]. It is sufficient to know how each of the coherent modes propagates, and eventually sum them incoherently.
In the case of a thin annular source, the expression of the ith mode across the source plane, say 9i(⇢, 0), can be obtained from equation (26) adding the appropriate dependence on the radial variable. This leads to
9i(⇢, 0)=p
K (⇢ a)s(') XN
j=1
c(i)j p
njeinj'. (30) The corresponding field, propagated in paraxial condi- tions at a distance z from the source plane, is evaluated by means of the Fresnel diffraction formula [28], which, in radial coordinates, reads
9i(r, z) = ikeikz
2⇡z exp✓ ikr2 2z
◆ Z 2⇡
0
Z 1
0 9i(⇢, 0)
⇥ exp⇢ ik
2z [⇢2 2r⇢ cos(# ')] ⇢ d⇢ d', (31) where r = (r, #) is a typical point across the arrival plane. On substituting from equation (30) into equation (31) we obtain
9i(r, z) = ikap Keikz
2⇡z exp ik(r2+ a2) 2z
XN j=1
c(i)j p
nj
⇥ Z 2⇡
0 s(')exp
inj' ikar
z cos(# ') d'. (32) To solve the latter integral, the explicit form of the function s(') must be given. In any case, analytical solutions can be obtained by resorting to the Fourier expansion of s('), which is a periodic function of its argument, i.e.
s(') = X1
m= 1
⌧meim', (33)
with ⌧msuitable coefficients. In such a way, each term of the Fourier series gives rise to a contribution that can be evaluated analytically, and the propagated field turns out to be
9i(r, z) = ikap Keikz
z exp ik(r2+ a2) 2z
⇥ XN
j=1
X1 m= 1
c(i)j p
nj⌧m( i)nj+m
⇥ Jnj+m✓ kar z
◆
ei(nj+m)#, (34)
where Jn(t)is the Bessel function of the first kind and order n and the following integral relation has been used [29]
Jn(t) = 1 2⇡
Z 2⇡
0 exp[i(n" t sin ")] d". (35) Of particular interest is the case in which the field radiated from the source is collimated by a converging lens, placed at a distance from the source equal to its focal length, say f . If the transverse limitations imposed by the finite aperture of the lens can be neglected, the field emerging from the lens, denoted by Zi(r), presents propagation-invariant coherence characteristics [18]. In fact, across the exit plane of the lens each of the modes has the form
Zi(r) = i p
Keikfexp✓ i a 2
◆XN j=1
X1 m= 1
c(i)j p
nj
⇥⌧m( i)nj+mJnj+m( r)ei(nj+m)#, (36) with = ka/f , which corresponds to a sum of Bessel beams of higher order. Since all such Bessel beams suffer the same dephasing upon propagation after the lens, namely, kzzwith kz = kq
1 (a/f )2, the only propagation effect on the mode structure is an overall phase change.
5. An example
5.1. Modal expansion
To present a simple application of the results of the previous sections, we consider a case that can be easily solved in an analytical way. We consider a degree of coherence of the form g('12)= cos '12, (37) with '12= '1 '2.
The only non-vanishing Fourier coefficients of this function are 1 and 1, both equal to 1/2. Since such coefficients are non-negative, the form in equation (37) may represent a bona fide degree of coherence. Furthermore, we know at once that the partially coherent source admits only two coherent modes, regardless of the specific form of s(').
It should also be noted that, from the expression of the matrix elements in equation (29), it follows that the matrix ˆA is diagonal whenever the Fourier coefficient of the intensity distribution 2 (and hence 2) is zero. When this happens, the functions U 1and U1in equation (14) are orthogonal and
4
Figure 1. Intensity distributions corresponding to the two modes in equation (46) for different values of the parameter ⌘.
coincide with the modes of the source, and the coefficients 1 and 2are its eigenvalues.
Let us then consider an intensity distribution for which
2 does not vanish. We take, for instance, a sinusoidally modulated intensity with contrast ↵ (0 ↵ 1), of the form
I(') = I0[1 + ↵ cos(2')] , (38) with I0a constant positive factor, whose only non-zero Fourier coefficients are
0= I0; +2= 2=↵I0
2 . (39)
The associated ˆA matrix is evaluated from equations (29) and (39) as
ˆA =⇡I0
2
"
2 ↵
↵ 2
#
(40)
which the following eigenvalues correspond to:
1= ⇡I0
⇣1 ↵ 2
⌘
, 2= ⇡I0
⇣1 +↵ 2
⌘
, (41) with eigenvectors
c1= 1 p2
"
1 1
#
, c2= 1
p2
"
1 1
#
. (42)
Note that adding Fourier components to the function in equation (38) does not affect the result in equations (41) and (42).
In conclusion, the eigenvalues of the source coincide with those in equation (41) and, from equations (14), (17), (20) and
(42), the corresponding modes turn out to be 81(')= s(')sin '
p(1 ↵/2)⇡I0, 82(')= s(')cos '
p(1 + ↵/2)⇡I0
,
(43)
where normalization factors have been included.
5.2. Propagation
The knowledge of the modal structure of the source can be used to evaluate the field radiated from it, following the procedure outlined in section4. To this aim, however, we have now to specify the analytical form of the modulation function s('). Up to now, in fact, we have used only the expression of I('), which specifies the square modulus of the function s('), while the latter is in general a complex function. To go ahead with a simple case, we choose
s(') = s0(ei'+ ⌘e i'), (44) with s0 and ⌘ constants (⌘ 0), which gives rise to the intensity in equation (38), with
I0= |s0|2(1 + ⌘2); ↵= 2⌘
1 + ⌘2. (45) In such a case, the only non-vanishing Fourier coefficients of s(') are ⌧1= s0 and ⌧ 1 = ⌘s0. Proceeding as specified in section 4, the two modes, propagated and collimated by a suitable converging lens, turn out to be proportional to the following functions (see equation (36)):
F1(r; ⌘) = (1 ⌘)J0( r)+ (e2i# ⌘e 2i#)J2( r); F2(r; ⌘) = (1 + ⌘)J0( r) (e2i#+ ⌘e 2i#)J2( r) .
(46)
J. Opt. 14 (2012) 035701 M Santarsiero et al
Figure 2. Transverse intensity distribution of the propagated field (from equation (49)) for different values of the parameter ⌘.
Plots of the squared modulus of the above functions are shown in figure1for ⌘ = 0, 0.5, 1.
In particular, when ⌘ = 0 (i.e. ↵ = 0) the intensity is uniform along the annulus and the modes become proportional to
F1(r; 0) = J0( r) + e2i#J2( r);
F2(r; 0) = J0( r) e2i#J2( r), (47) while in the opposite limit of maximum intensity modulation (⌘ = 1, i.e. ↵ = 1) they become
F1(r; 1) = sin(2#)J2( r);
F2(r; 1) = J0✓ kar f
◆
cos(2#)J2( r). (48) The mutual intensity of the field after the lens, say JL, can be then evaluated from the knowledge of the modes (equation (46)) and of the eigenvalues (equation (41)), and turns out to be
JL(r1,r2)= AI0
1 + ⌘2[F1(r1; ⌘)F1⇤(r2; ⌘)
+ F2(r1; ⌘)F⇤2(r2; ⌘)], (49) where in the coefficient A all the constant factors have been included. In particular, the transverse intensity distribution can be evaluated from equation (49) on letting r1= r2(see figure2).
6. Conclusions
Thin annular sources of either coherent [16, 17] or incoherent [10] nature were of utmost importance in the generation process of diffraction-free and J0-correlated beams, respectively. In a more general case, thin annular sources with partial coherence were considered [15]. A common feature of all these partially coherent sources was a uniform intensity distribution along the annulus. Here, we extended previous treatments in order to remove this limitation and discussed the modal analysis in this general case and its use in the study of propagation phenomena.
In such a way, more general propagation-invariant partially coherent beams can be studied and, by resorting to incoherent superposition schemes of the coherent modes, further synthesis procedures can be envisaged for this type of field structure.
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