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Rotating correlations inpartially coherent fieldsToni Saastamoinen a & Jani Tervo aa University of Joensuu, Department of Physics, POBox 111, FIN-80101, Joensuu, FinlandPublished online: 03 Jul 2009.

To cite this article: Toni Saastamoinen & Jani Tervo (2004): Rotating correlations inpartially coherent fields, Journal of Modern Optics, 51:5, 633-643

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JOUIINAI, OF R.IOI)I:.RN OPTICS, 20 MARCH 2004 \ . ( ) I , . 51 , YO. 5 , 633-643

+ Taylor &Francis 0 Taylor 6Francir Group

Rotating correlations in partially coherent fields

T O N I S A A S T A M O I N E N and JAN1 TERVO* University of Joensuu, Department of Physics, PO Box 1 1 1 , FIN-80101 Joensuu, Finland

(Received 2 7 May 2003)

Abstract. Partially coherent optical fields whose cross-spectral density func- tions rotate on propagation are examined. The general theory for rotating partially coherent fields in the space-frequency domain is derived for both scalar and electromagnetic approaches. Differences between the results obtained with full and partial coherence is discussed. A numerical example is given for rotating intensity distributions.

1. Introduction T h e interest in the electromagnetic approach to partial coherence has grown

rapidly during the last few years [l-91. Although the tools for the rigorous treatment of electromagnetic partially coherent fields were derived over 40 years ago [ 10-1 21, recent research has been focusing mainly on paraxial vector fields. This is probably because the non-paraxial theory requires simultaneous examina- tion of 36 correlation functions instead of only four required in the paraxial case. However, the formal complexity of the full electromagnetic analysis of partially coherent fields may be overcome with the help of Maxwell's equations which strictly connect the correlation functions. In fact, in free-space propagation it is sufficient t o consider only three correlation functions and the remaining 33 unknown correlation functions may be solved from Maxwell's equations [13].

In this article, we apply the angular spectrum representation for partially coherent wave fields to find out the propagation laws for partially coherent fields whose correlation properties remain invariant on propagation, except for the rotation which is linearly dependent on the propagation distance. T h i s class of fields is closely connected to the propagation-invariant, that is non-diffracting, fields put forward by Durnin [14] in 1987 and Durnin et al. [15] and later extended to the domain of partial coherence by Turunen et al. [16].

T h e conditions for rotation for fully coherent fields are well known in both the scalar [17-201 and the electromagnetic [21, 221 approaches. However, it is not known how partial coherence affects the behaviour of this particular class of opti- cal fields. Although the concepts of rotation are similar in both fully and partially coherent cases, there is a fundamental difference between these two classes. Namely, in the analysis of coherent fields we analyse directly the behaviour of wave fields but in the partially coherent case we must gather all the information by examining the correlation functions. Th i s often has implications for results and their interpretation.

"Author for correspondence; E-mail: jani.tervo($,joensuu.fi

Jorrrrml ~f ,\lodiw Oprirr l S S N 09j*0340 printiISSN 1362-3044 online ( I ' 2004 'Taylor & Francis L,td http://wwu.tandf.co.uk/journals

DOI: io.1o~n~o95oo340~ inn01 025777

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634 T Saastamoinen and J. Tervo

T h e paper is arranged as follows: In section 2 we start with the basics of the theory of partial coherence in the space-frequency domain and briefly recall the angular spectrum representation of wave fields. In section 3 we derive the expressions for rotating scalar fields and extend the analysis to the electromagnetic domain in section 4. T h e conclusions are made in section 5 .

2. Angular-spectrum representation One of the central results of Wolf's [23, 241 theory of partial coherence in the

space-frequency domain is the fact that we may express the cross-spectral density function W of a statistically stationary field at points rl and r2 as an ensemble average:

W(rl ,r2; w ) = (U*(rl; w)U(r2; w ) ) , (1) where U(r; w ) denotes a monochromatic scalar function of frequency w and the asterisk denotes complex conjugation. Since W is a correlation function, it must fulfil the Hermiticity relation [25]

W(rl; r2; w ) = W*(r2; rl ; w ) (2)

as well as the non-negative definiteness condition

where f l and f2 are arbitrary, sufficiently well-behaving functions and the domain of integration is arbitrary.

If we assume that the half-space z>O is free of sources, we may apply the angular spectrum representation of scalar wave-fields to each realization U(r; w ) . This yields the following expression for the cross-spectral density function [25]:

W(r1; r2; w ) = / / $ / r d ( k l l ; k21; 0) exp [jam ~0442 - @2)1 exp [-ialpl

x cos(41 - @I)] exp [ i ( k 2 ~ 2 - k;,zi)la1a2 d@l dlCI2 daidaz, (4) 0

where we have chosen to express both the position vectors and the wave vectors k,, j = 1, 2, in circular cylindrical coordinates, that is

kjx = aj cos @j,

kjS = aj sin @j, and { xj = pj cos 4j

yj = pj sin 4j

and employed the short-hand notation k j l = (kj,., kjy) for the transverse part of the wave vectors. T h e z components of the wave vectors are obtained from the relation

where k = llk,ll = w / c and c is the speed of light in vacuum. T h e real-valued solutions of equation (6) correspond to the propagating waves, whereas the imaginary solutions correspond to the evanescent waves whose amplitudes decrease exponentially as z increases. T h e function d ( k l l ; k21; w ) is called the angular correlation function and it is defined as an ensemble average over

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Rotating correlations in partially coherent fields 635

the plane-wave components A(k,, k,,; w ) of the field realizations. Equation (4) indicates that the angular correlation function is obtained by Fourier inversion at zl = z2 = 0:

If the electromagnetic nature of the field is taken into account, a single scalar- valued cross-spectral density function is incapable of describing the properties of the field. In this case the correlations must be examined by using the four 3 x 3 cross-spectral density tensors describing the correlations between the components of the electric and the magnetic fields [25]. However, in section 4 in which the electromagnetic solutions are concerned, we are restricted to considering the properties of the electric field E only, since the magnetic field may be obtained with the help of Maxwell’s equations. Thus it is sufficient to study the correlations defined only by the electric cross-spectral density tensor

where i , j = x , y , z. In what follows, the explicit use of the subscript or superscript e denoting the electric field is omitted, since no other matrices than the electric cross-spectral density matrix are examined.

The Hermiticity and the non-negative definiteness conditions corresponding to equations (2) and (3) now take the forms [25]

and

respectively. The angular spectrum representation may be used also in the electromagnetic

case such that equations (4) and (7) are applied separately for each element of the cross-spectral density matrix [ 13, 261. However, now Maxwell’s divergence equation V . E = 0 states that the direction of the electric-field vector of each plane-wave component must be perpendicular to the corresponding wave vector. This, together with equation (9), leads to the fact that only three of the nine elements of the cross-spectral density matrix are needed to determine fully the behaviour of the field [13]. Because of the chosen geometry, we choose the independent components to be W,,, W,, and WYy.

3. Rotating scalar correlations In the scalar theory of rotating fully coherent fields [17-211 the definition of

rotation was made by assuming that the intensity distributions in two planes are identical, except for the rotation of an angle linearly dependent on the distance between the planes. This condition may be expressed in mathematical form by

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stating that [21]

I U(P, 4 + u Az, z + Az; w)I2 = I U(P, 4, z ; w)I2, (1 1)

where v is a real constant defining the direction, as well as the amount, of rotation after propagation of a certain distance denoted by Az. Condition (1 1) leads to the requirement that the angular spectrum must be confined on a discrete set of concentric circles, known as Montgomery's [27, 281 rings. This result originates from the fact that a rotating field is also self-imaging, the transverse intensity distribution being repeated after rotating 2x rad, or its integer fractions. In addition to the self-imaging condition, a rotating scalar field must be formed by a superposition of propagation-invariant fields of strictly limited orders. Thus the field may be expressed in the form [21]

u( P, 4, z ; w ) = C am(@) Jm(anzP) ~ X P [i(m4 + kzrnz)~, (1 2) mGM

where a,(w) are complex amplitudes, J, denote the Bessel functions of orders m, and a$ + k;, = k2. The orders m and the radii a,,, of Montgomery's rings are connected by the equation

k,, = ko - mv,

where ko is a constant. The symbol M appearing in equation (12) denotes the set of permitted orders m, that is for which 0 < k,, < k.

In the case of partial coherence the rotation condition may be defined in several different ways. One possibility is to demand, as in the case of full coherence, that the intensity distribution I(r; w ) = W ( r ; r; w ) rotates. On the other hand, one could require that the cross-spectral density function itself rotates, that is that

(1 3)

W( pi, 41 + v Ax, ZI + AZ ~ 2 , 4 2 + Az, z2 + Az; w ) = PI, 41, ZI ; ~ 2 ~ 4 2 , z2 ; W )

(14)

holds for all A z > 0. The latter definition is more severe since, by choosing rl = r2 in equation (14), we find that the intensity distribution is rotating. Because of that we shall examine here the solutions of equation (14) only.

T o attack the problem we first apply the Jacobi-Anger expansion [29] m

exp(iy cos e) = C 1 .m J , ( y ) exp(im8) (1 5)

to equation (4) and then interchange the order of summation and integration. This yields

m=-m

Since the angular correlation function is periodic in the azimuthal direction, it may be expanded as a Fourier series

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Rotating correlations in partially coherent jields 637

with the functions d,,, ,, given by

By comparing equations (16) and (18) we find that the cross-spectral density function may be expressed in the form

x ~ m ( a l p l ) ~ n ( ~ p 2 ) exp [i(k2,z2 - k ; , z i ) ] a i ~ dai da2. (19)

It should be noted that although the functional form of equation (19) differs rather radically from equation (4), they are equivalent, that is, any assumption of rotation has not yet been made.

Let us now insert equation (19) into equation (14), from which we obtain the condition

x e ~ p [ i ( k 2 ~ ~ 2 - k ~ , z l ) ] { e x p [ i ( n ~ - r n u + k 2 , - k ~ , ) A z ] - l}ala2dal da2 ~ 0 . (20)

Equation (20) may hold only in the case that the integral vanishes for all combinations of m and n. On the other hand, since equation (20) must hold for all z1, z2 > 0, we find that the integrand itself is identically zero. This means that

exp[i(nu - mu + k2, - kr,)Az] = 1 (21)

must be true for all RnJal; a2; w ) # 0. Equation (21) implies that both k l , and kzZ are real valued which is, of course, natural because no evanescent waves may be a part of a rotating field. Since A z in equation (21) may be chosen arbitrarily, we have

( n - m)u + k2, - k l , = 0 (22)

for d,,,,,, # 0 and, consequently,

and 2) denotes the set of allowed values of a and which are restricted to the interval [O,k] . Equation (24) gives the general solution of the condition (14). However, unlike in the fully coherent case, the choice of functions dtn, n(a; w ) is not

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638 T Saastamoinen and J. Tervo

free, since d(kl1; k21; w ) is a correlation function and therefore it must satisfy the Hermiticity relation

d * ( k u ; k i i ; w ) = d(ki1; k21; w ) (26)

and the non-negative definiteness condition corresponding to equation (3) . By inserting equation (17) into equation (26) and then using equations (23) and (25), we obtain the rule

d:,m(am,n; w ) = dm,n(a; w). (27)

T o obtain some physical insight to the rotation condition, we express an arbitrary scalar wave function U(r; o) propagating in the positive z direction in the form

00

U(r; w ) = 2rc i" exp (imq5) &(a; w)J,Jap) exp (ik,z)a da, (28) m=-00 1:

2n s, where

(29) 1 2n

Am(a; w ) = - A(k1; o) exp (-im@) d@.

Thus, U may be understood to be formed as a superposition of non-diffracting subfields

Urn,Jr; w) = 2rci"' exp (im#)A,(a; w ) J , (ap ) exp (ik,z), (30)

whose mutual relations depend on the complex-amplitude functions A,(a; w). The angular spectrum of each subfield is confined on a single Montgomery's ring with radius a. By comparing equations (16) and (28), we find that we may make an interpretation

Am,n(al; ~ 2 ; w) = ( A ; ( ~ I ; ~)An(a2; w)). (31)

If we now examine equations (23) and (31) , we find that the functions Am(al; w ) and An(a2; w), and hence also the corresponding subfields, may correlate only if the difference between the radii a1 and cq of Montgomery's rings of a certain value determined by equation (22).

A numerical example of a rotating partially coherent scalar field is illustrated in figure 1. The field is calculated with the parameters

where C(w) is an intensity-scaling constant, k,, = 99k/100 and kb, = 96k/100. The example is chosen such that the total field is a superposition of three non-diffracting fields (with random amplitudes). The amplitudes of two of the subfields are mutually partially correlated whereas the third is uncorrelated with the other two.

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Rotating correlations in partially coherent fields

Z = O Z = zT/4

639

Figure 1. T h e intensity distribution of a rotating partially coherent scalar field as a function of distance z within one self-imaging distance z-,..

4. Electromagnet ic solutions In the analysis of fully coherent rotating fields [21] it was found that the

(longitudinal) z component of a field, whose (transverse) x and y components rotate, is not generally rotating. This clearly means that the intensity distribution ( i x . electric energy density) does not rotate. The phenomenon arises from the fact that the propagated version of such a field is not a rotated replica of the original in a vectorial sense [21]. However, if the radial and azimuthal components of the field, defined by

and

E@(r; w ) = - sin@E,(r; w ) + cos 4EJr; w ) , (34)

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are assumed to be rotating, then also the z component rotates owing to the vectorial rotational symmetry. This leads to a rotation condition slightly different from equation (13):

kzm = KO - (m + s ) ~ , (35) where s may assume values f l .

Because of the differences between the scalar and electromagnetic solutions in the fully coherent case, one could expect that the solutions differ also if the rotating correlations are concerned. However, since no actual fields can be examined in the partially coherent case, we cannot use the symmetry reasons as a starting point, as was done in the study of coherent fields [21]. Instead, we require that each element of the 2 x 2 matrix

rotates, that is

In other words, we require that the (transverse) cylindrical correlation functions rotate exactly in the same way.

By using equations (19), (33) and (34), we obtain the expressions for the cross- spectral density functions describing the correlations of the transverse cylindrical components, in the matrix form

M M

~ ( r l ; r2; W ) = (2x1~ C C ill-” exp [i(n42 - m41>] m = - w n = - w

x //rR(4I)Arn,n(ai; a2; w ) R ( - ~ ~ ) J ~ ~ ( ~ I P I ) J ~ ( ~ ~ P z )

where

and R(4) is the usual rotation matrix defined by

C O S ~ s in4 (40)

By inserting equation (38) into equation (37), we obtain, after a few steps, the following system of equations

dz ,,Frnn + dz: Gmn + d”x m , n Hmn + dyy m , n Kmn = 0 ,

,,Grn,i + A;: .Finn -

$rnn - A:: &mn + .Kinit + A;: ,,Hrnn = 0 ,

,,Frnn + A;{ ,,Grnn = 0 ,

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Rotating correlations in partially coherent fields 64 1

where d~,,ll = d;,,,(al; a2; w),

Frnn = cos (41 + uAz) cos (42 + uAz) exp (ion,,, Az) - cos cos 42, Gvlll cos (41 + uAz) sin (42 + uAz) exp (iOmll Az) - cos 41 sin 42,

(42) Hl1,,,

K,,,

= sin (41 + uAs) cos (42 + uAz) exp (i&ll Az) - sin 41 cos 42,

= sin (41 + VAG) sin (42 + uAz) exp (iOlllTl Az) - sin 41 sin 42,

and ~ n l , l = ( n - m ) u + k2z - kl,. The solution of system (41) is found by requiring that the determinant of the coefficient matrix vanishes, that is

I Fmn Gmn H m n Krnn

det[ - G m n F m n -Kmn Hint,

-Hmn -Kmi Fmii Gmn K i n n -Hmn -Ginn Fnin

Equation (43) implies that equation (41) has four solutions which connect the angular correlation functions and radii of the Montgomery’s rings. The solutions can be divided into two classes:

d , ~ , ~ , , ( a ~ ; a 2 ; w ) = d ~ ~ , , ( l r ~ ; a 2 ; w ) = ~ i d ~ ~ , , ( a ~ ; a 2 : w ) = ~ i d ~ ~ ~ , , ( a ~ ; a 2 ; w ) , k2z - k l , + ( n - m)u = 0,

(44)

and

dzll(al; az; w ) = -dL<n(al; a2; w ) = &id;<,,(al; a2; w ) = *id:;,,(al; a2; w), k l z - k l , + ( n - m - 2)u = 0.

(45)

It is evident that we can ignore the latter because it does not satisfy the requirements for correlation functions nor the Hermiticity condition. However, the first is valid and it indicates that the field consists of either left-handed or right-handed, circularly polarized components.

By comparing equations (22) and (44) we find that the rule connecting the wave vectors are equivalent in both scalar and electromagnetic cases. This might sound surprising, especially if we recall that in the case of fully coherent fields the corresponding conditions are different for scalar and electromagnetic fields [21]. The obtained result arises from the fact that in the coherence theory we are examining correlation functions instead of actual fields. Whereas the correspond- ing rule for the fully coherent case gives the permitted orders of the Bessel field modes, equations (22) and (44) describe how different modes may correlate, as was discussed in the previous section.

According to equation (44), the sign of d;:, can be either positive or negative for every ( m , n ) pair. By combining equation (44) with equations (38)-(40) and

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(25), and rearranging the indices in the summations, we obtain

W(r1; r2; w ) = ( 2 ~ ) ~ c 1 exp[i(n42 - m41)I PsJm+s(apl) ' m o o

m = - w II=--M s=- l , 1 I D

where

and D is the same interval as in the scalar analysis. The remaining unknown elements of the cross-spectral density tensor are

obtained by using the Hermiticity relation (9) and the Maxwell's divergence equation in cylindrical coordinates. This yields

m=-w n=-m s=-l,l

x exp [ik,(z2 - zl)] exp [i(m - n)vzz] aam, n da , (49)

x exp [ikz(z2 - zl)] exp [i(m - n)uz2] aa,, ,, da , (50)

where amn is defined by equation (25) and k,,m,n = k, + (m - n)u. Equation (46) and (48)-(SO) and the remaining two components (which may be

obtained with the help of the Hermiticity relation (9)) are the general solutions to the condition (14) for the electric field. The elements of the magnetic and the mixed cross-spectral density matrices can be derived straightforwardly by apply- ing Maxwell's equations. It is easy to show that all the elements of the mentioned matrices do rotate, provided that the cylindrical components of the field are examined.

5. Conclusions In this paper we have extended the concept of rotating optical fields into the

domain of partial coherence and partial polarization. The explicit expressions were presented for the cross-spectral density functions and the conditions for rotation in

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Rotating correlations in partially coherent fields 643

both scalar and electromagnetic cases. It was shown that the rule connecting the orders of Bessel functions and Montgomery's rings are exactly the same in both scalar and electromagnetic treatments, although in the analysis of coherent fields the corresponding rules are known to differ. This result arises from the fact that in the coherent optics the rotation condition fixes the number of allowed field modes in each Montgomery's ring whereas in the case of partial coherence the condition determines how the different modes correlate with each other.

Acknowledgments T. Saastamoinen wishes to thank the Finnish Academy of Science and Letters

for financial support. The work of J. Tervo is supported by the Academy of Finland (project 201008).

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