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368 J. Opt. Soc. Am. A/Vol. 24, No. 2 /February 2007 Liu et al.

Propagation theories of arbitrary-ordercorrelations of partially coherent electromagnetic

fields based on separated-coordinatemode decomposition

Haitao Liu, Guoguang Mu, and Lie Lin

Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education,Institute of Modern Optics, Nankai University, Tianjin 300071, China

Received May 5, 2006; revised July 30, 2006; accepted August 30, 2006;posted September 1, 2006 (Doc. ID 70565); published January 10, 2007

Theories to calculate the propagation of arbitrary-order correlations of stationary or nonstationary partiallycoherent electromagnetic fields are proposed. The theories are based on separated-coordinate mode decompo-sition, and can make the well-developed propagation theories of fully coherent electromagnetic fields appli-cable to partially coherent electromagnetic fields governed by linear Maxwell equations. The validity of thetheories is illustrated by an example. © 2007 Optical Society of America

OCIS codes: 030.6600, 030.4070, 260.2110.

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. INTRODUCTIONheories for the propagation of fully coherent light haveeen developed sufficiently, for example, the geometricalptics1 and the rigorous diffraction theories.1,2 In com-arison, propagation theories of partially coherent lightave not been established as well. A possible solution iseported in Refs. 3 and 4, which is to make the propaga-ion theories of fully coherent light applicable to theropagation of partially coherent light, based on coherentode decomposition5–8 (CMD) or the separated-

oordinate mode decomposition (SCMD).4 In CMD theerms are physical modes that add incoherently. In SCMDach mode takes a form with the spatial coordinates sepa-ated. One merit of SCMD is that it can avoid solving theigenvalue integral equations of CMD, but at the cost ofore modes than CMD. Compared with CMD, SCMD is

ather a mathematical concept than a physical conceptince each mode in SCMD may not be a physical mode. Inef. 3 the propagation of the second-order correlations oftationary partially coherent scalar fields based on CMDs addressed. The propagation of the second-order corre-ations of stationary partially coherent electromagneticelds based on CMD or SCMD is discussed in Ref. 4.However, higher-order correlations of partially coher-

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nt light have not only theoretical merits but also appli-ation significance,5 such as in the classical analysis of in-ensity interferometer.9 In addition, nonstationary fieldsuch as the partially coherent pulses are attracting in-reasing interest.10,11 In this paper, we propose the propa-ation theories of arbitrary-order correlations of station-ry or nonstationary partially coherent electromagneticelds based on SCMD. The proposed theories are an ex-ension of the theories in Ref. 4, which can make theropagation theories of fully coherent electromagneticelds applicable to the propagation of arbitrary-order cor-elations of partially coherent electromagnetic fields gov-rned by linear Maxwell equations.

We organize this paper as follows: the propagationheories are presented in Section 2, an example is pro-ided in Section 3, and conclusions are summarized inection 4.

. PROPAGATION THEORIES. Definitions of Arbitrary-Order Correlationse define a general arbitrary-order correlation tensor of

he nonstationary partially coherent electromagneticelds in the space-time domain as5

�K,L,M,N��r1,r2, . . . ,rK+L+M+N;t1,t2, . . . ,tK+L+M+N� =��k=1

K

e�rk,tk� �l=K+1

K+L

h�rl,tl� �m=K+L+1

K+L+M

e*�rm,tm� �n=K+L+M+1

K+L+M+N

h*�rn,tn�� , �1a�

here r is the spatial coordinate, t is time, e�r , t� and�r , t� are the analytic signals of the random electric andagnetic vectors in the space-time domain, respectively,

uperscript * means complex conjugate, � � means the en-emble average, and � means the exterior product of vec-ors, i.e.,

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Liu et al. Vol. 24, No. 2 /February 2007 /J. Opt. Soc. Am. A 369

�k=1

K

e�rk,tk� = e�r1,t1�e�r2,t2� ¯ e�rK,tK�, �1b�

hich is a Kth-order tensor. �J�K,L,M,N� defined in Eq. (1a)s a �K+L+M+N�th-order tensor. It is notable that the or-er of the factors on the right side of Eqs. (1) cannot behanged. The spectrums of e�r , t� and h�r , t� are

E�r,�� =��

e�r,t�exp�i2��t�dt,
−� s
BNTtm

wv�pptmi=Hcfias

cenSdfi

H�r,�� =�−�

�

h�r,t�exp�i2��t�dt, �2�

here � is frequency, and E�r ,�� and H�r ,�� are the ran-om electric and magnetic vectors in the space-frequencyomain, respectively, which are of zero value if ��0. Aeneral arbitrary-order correlation tensor of the nonsta-ionary partially coherent electomagnetic fields in the

5
pace-frequency domain can be defined as
�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N� =��k=1

K

E�rk,�k� �l=K+1

K+L

H�rl,�l� �m=K+L+1

K+L+M

E*�rm,�m� �n=K+L+M+1

K+L+M+N

H*�rn,�n�� . �3�

y Eqs. (2), the space-time correlation tensor and thepace-frequency correlation tensor are connected by theourier transform of

�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N�

=�−�

�

dt1�−�

�

dt2 ¯�−�

�

dtK+L+M+N�J�K,L,M,N�

��r1,r2, . . . ,rK+L+M+N;t1,t2, . . . ,tK+L+M+N�

��k=1

K+L

exp�i2��ktk� �m=K+L+1

K+L+M+N

exp�− i2��mtm�, �4a�

here � means product of scalars, i.e.,

�k=1

+L

exp�i2��ktk�

= exp�i2��1t1�exp�i2��2t2� ¯ exp�i2��K+LtK+L�. �4b�

or stationary fields, the space-time correlation tensor isndependent of the origin of time,5 namely,

�K,L,M,N��r1,r2, . . . ,rK+L+M+N;t1,t2, . . . ,tK+L+M+N�

= �J�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�,

�5�

here �i= ti− t1, i=2,3, . . . ,K+L+M+N. The insertion ofq. (5) into Eq. (4a) can yield5

WJ�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N�

= ��1 + �2 + ¯ + �K+L − �K+L+1 − �K+L+2 − ¯

− �K+L+M+N� � VJ�K,L,M,N�

��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�, �6a�

here � is the Dirac delta function and

VJ�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�

=�−�

�

d�2�−�

�

d�3 ¯�−�

�

d�K+L+M+N�J�K,N,M,N�

��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�

� �k=2

K+L

exp�i2��k�k� �m=K+L+1

K+L+M+N

exp�− i2��m�m�. �6b�

. Presentation of Propagation Problem ofonstationary Fieldshe Maxwell equations in the space-frequency domain

hat governs the linear propagation of random electro-agnetic fields take a linear form of1

� � E�r,�� = i2��r,��H�r,��,

� � H�r,�� = − i2��r,��E�r,��, i.c. E��i��,��, �7�

here �=x�� /�x�+y�� /�y�+z�� /�z�, x ,y ,z are the unitectors along the three coordinate axes, r=xx+yy+zz,�r ,�� and �r ,�� are the magnetic conductivity and theermittivity of the medium, respectively, which are de-endent on r and � in general, “i.c. E�

�i�� ,��” means thathe transverse component E�

�i�� ,�� of the incident or illu-inating random electric field on plane z=0 acts as the

llumination condition, E��i�� ,��=xEx

�i�� ,��+yEy�i�� ,��, �

xx+yy is the coordinate on plane z=0, and E�r ,�� and�r ,�� can denote the electric and magnetic vectors of in-

ident, scattered or total fields since the three kinds ofelds all satisfy the Maxwell equations (7). The incidentnd scattered fields are within the same region, whoseum is the total field in this region.

We take some examples to illustrate the concepts of in-ident, scattered, and total fields mentioned above. Onexample is the reflection and refraction of fields on a pla-ar dielectric interface located at z=d, as presented inection 3. Within the region of z�d, the sum of the inci-ent field and the scattered (reflected) field is the totaleld; within the region z�d, the transmission field is just

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he total field. The Maxwell equations (7) are satisfied byhe incident and scattered fields within z�d and by theotal field within the whole space. Another example is theell-known scattering of an incident planar wave by a di-lectric sphere.1 Out of the dielectric sphere, the sum ofhe incident planar wave and the scattered field is the to-al field; within the dielectric sphere, only the total field issually considered without being decomposed into the in-ident planar wave and the scattered wave. The Maxwellquations (7) are satisfied by the incident and scatteredaves out of the dielectric sphere and by the total field in

he whole space. A third example is the diffraction ofelds in homogenous half space z�0, as shown by Eqs.9). Within the half space z�0, there is no scattered fieldnd the incident field is just the total field that satisfieshe Maxwell equations.

By physical intuition, if the medium geometry of ��r ,��nd �r ,�� is assumed to be known, the Maxwell equa-ions (7) should have a unique solution dependent on thencident field and therefore dependent on E�

�i�� ,��, be-ause both the electric and the magnetic vectors of the in-ident field can be uniquely determined by E�

�i�� ,�� ashown in Eqs. (9) if we assume that there is no incidenteld from infinity in z�0.12 Based on this point, the Max-ell equations (7) can be rewritten with mathematicalquivalence in a solution form4

E�r,�� = PJ�e��r,�,�� · E��i��,��,

H�r,�� = PJ�h��r,�,�� · E��i��,��. �8�

here PJ�e��r ,� ,�� and PJ�h��r ,� ,�� are two abstract opera-
ors. By using the linear property of the Maxwell equa- f
at

ions (7), it has been proved in Ref. 4 that PJ�e��r ,� ,�� and�h��r ,� ,�� are both 3�2 tensor operators with all their

ix elements being linear operators, and · in Eqs. (8)eans inner product. For example, in homogenous half

pace z�0, the Maxwell equations (7) are equivalento4,13,14

Ex�r,�� = −1

2��

−�

� �G�r,�,��

�zEx

�i��,��d2�,

Ey�r,�� = −1

2��

−�

� �G�r,�,��

�zEy

�i��,��d2�,

Ez�r,�� =1

2��

−�

� � �G�r,�,��

�xEx

�i��,�� +�G�r,�,��

�y

�Ey�i��,��d2�,

H�r,�� = �i2��−1 � � E�r,��, �9�

here E�r ,��=xEx�r ,��+yEy�r ,��+zEz�r ,��, G�r ,� ,��exp�ik r−� � / r−� , k=2�� /c, c is the light velocity in

he half space, and PJ�e��r ,� ,�� and PJ�h��r ,� ,�� take the
orm
PJ�e��r,�,�� = �x,y,z��−

1

2��

−�

�

d2��G�r,�,��

�z0

0 −1

2��

−�

�

d2��G�r,�,��

�z

1

2��

−�

�

d2��G�r,�,��

�x

1

2��

−�

�

d2��G�r,�,��

�y

��x

y� ,

PJ�h��r,�,�� =1

i2��x,y,z��

1

2��

−�

�

d2��2G�r,�,��

�x � y

1

2��

−�

�

d2�� 2G�r,�,��

�y2 +�2G�r,�,��

�z2 �−

1

2��

−�

�

d2�� 2G�r,�,��

�x2 +�2G�r,�,��

�z2 � −1

2��

−�

�

d2��2G�r,�,��

�x � y

1

2��

−�

�

d2��2G�r,�,��

�y � z−

1

2��

−�

�

d2��2G�r,�,��

�x � z

��x

y� .

�10�

Now we come back to the presentation of the propaga-ion problem of arbitrary-order correlations of nonstation-

ry partially coherent electromagnetic fields. The inser-ion of Eqs. (8) into Eq. (3) can yield

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=��k=1

K

�PJ�e��rk,�k,�k� · E��i��k,�k��

l=K+1

K+L

�PJ�h��rl,�l,�l�

· E��i��l,�l��

m=K+L+1

K+L+M

�PJ�e��rm,�m,�m�

· E��i��m,�m��* �

n=K+L+M+1

K+L+M+N

�PJ�h��rn,�n,�n� · E��i��n,�n��*� .

�11a�

o avoid the abstract operation of tensors, we rewrite Eq.11a) in the component form

W�1,�2,. . .,�K+L+M+N

�K,L,M,N� �r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N�

=��k=1

K

��

P�k �e� �rk,�k,�k�E

�i��k,�k��

l=K+1

K+L

��

P�l �h� �rl,�l,�l�E

�i��l,�l��

m=K+L+1

K+L+M

��

P�m �e� �rm,�m,�m�E

�i��m,�m��*

� �n=K+L+M+1

K+L+M+N

��

P�n �h� �rn,�n,�n�E

�i��n,�n��*� , �11b�

here �� x ,y ,z, �� x ,y (which are stipulatedhroughout the paper), and


= ��1,�2,. . .,�K+L+M+N

�1,�2, . . . ,�K+L+M+NW�1,�2,. . .,�K+L+M+N

�K,L,M,N�

��r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N�,

PJ�e��r,�,�� = ��,

��P� �e��r,�,��,

PJ�h��r,�,�� = ��,

��P� �h��r,�,��,

E��i��,�� = �

�E �i��,��. �11c�

s shown in Appendix A, by using the linear property of�e��r ,� ,��, PJ�h��r ,� ,��, and � �, Eq. (11b) can become

W�1,�2,. . .,�K+L+M+N


= � 1, 2,. . ., K+L+M+N

�k=1

K

P�k k

�e� �rk,�k,�k� �l=K+1

K+L

P�l l

�h� �rl,�l,�l�

� �m=K+L+1

K+L+M

P�m m

�e�* �rm,�m,�m� �n=K+L+M+1

K+L+M+N

P�n n

�h�* �rn,�n,�n�

� W 1, 2,. . ., K+L,M+N

�K+L,M+N��e��i�

��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�, �12a�

here

1, 2,. . ., K+L,M+N

�K+L,M+N��e��i� ��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�

=��k=1

K+L

E k

�i��k,�k� �m=K+L+1

K+L+M+N

E m

�i�*��m,�m�� . �12b�

in Eq. (12a) means the composition of operators, as ex-lained in Appendix A. We define a correlation tensor ofhe transverse incident electric field on the plane z=0 inpace-frequency domain as

WJ��K+L,M+N��e��i��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�

= � 1, 2,. . ., K+L+M+N

�1,�2, . . . ,�K+L,M+NW 1, 2,. . ., K+L+M+N


��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�, �12c�

hich is a �K+L+M+N� th-order tensor. By Eqs. (12), weome to an important conclusion that WJ�K,L,M,N�, satisfy-ng

K + L = K, M + N = M, �13�

an be uniquely determined by WJ��K,M��e��i�, while

�K,L,M,N� satisfying Eqs. (13) is not unique but has �K1��M+1� different cases. Now the propagation problemf arbitrary-order correlations of nonstationary partiallyoherent electromagnetic fields can be expressed as solv-ng WJ�K,L,M,N� defined in Eq. (3) with WJ�

�K+L,M+N��e��i� de-ned in Eqs. (12) as given.

. Definition of Fully Coherent Electromagnetic Fieldst is necessary to clarify the definition of fully coherentlectromagnetic fields, since the theories in this paper areo calculate the propagation of partially coherent electro-agnetic fields by using the propagation theories of fully

oherent electromagnetic fields.Our discussion is first addressed at the random trans-

erse incident electric field E��i�� ,�� on plane z=0 and

hen at the random propagation field E�r ,�� and H�r ,��,ince E�

�i�� ,�� can uniquely determine the propagationeld. By the CMD of partially coherent electromagneticelds,4,6,7,11 a reasonable definition of fully coherent elec-romagnetic fields is that the second order correlation ten-or WJ�

�1,1��e��i��1 ,�2 ;�1 ,�2�= �E��i��1 ,�1�E�

�i�*��2 ,�2�� con-ains only one coherent mode, namely,

WJ��1,1��e��i��1,�2;�1,�2� = E�C

�i� ��1,�1�E�C�i�*��2,�2�. �14�

quation (14) is for the case of nonstationary fields.11

or the case of stationary fields,4,6,7 in Eq. (14)

��1,1��e��i��1 ,�2 ;�1 ,�2� should be replaced by

��1,1��e��i��1 ,�2 ;�� defined in Eqs. (27) with �1=�2=�, and

he following discussions can be performed analogously.rom Eq. (14), it can be derived in Appendix B that theandom vector E�

�i�� ,�� and the definite vector function�i� �� ,�� are related by
�C

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E��i��,�� = E�C

�i� ��,��Ex�i��0,�0�,

Ex�i��0,�0� =

Ex�i��0,�0�

�� Ex�i��0,�0� 2�

exp�i�x�, �15�

here �0 and �0 are a fixed coordinate and a fixed fre-uency, respectively, which can be selected arbitrarily,nd �x is independent of � and �, which can be set anyeal value, so Ex

�i��0 ,�0� is a random variable independentf � and �. Then by the propagation Eqs. (8), the randomropagation field E�r ,�� and H�r ,�� can be determined as

E�r,�� = EC�r,��Ex�i��0,�0�, H�r,�� = HC�r,��Ex

�i��0,�0�,

�16a�

EC�r,�� = PJ�e��r,�,�� · E�C�i� ��,��,

HC�r,�� = PJ�h��r,�,�� · E�C�i� ��,��. �16b�

n view of the equivalence between Eqs. (8) and the Max-ell equations (7), Eqs. (16b) are equivalent to

� � EC�r,�� = i2��r,��HC�r,��,

� � HC�r,�� = − i2��r,��EC�r,��, i.c. E�C�i� ��,��.

�17�

ince the random variable Ex�i��0 ,�0� independent of �

nd � has no effect on the relative field distribution, Eqs.16a) and (17) show that the random propagation fields�r ,�� and H�r ,�� of fully coherent electromagnetic fields

an be determined by solving the Maxwell equations. Soqs. (16a) and the Maxwell equations (17) can act as an-ther definition of fully coherent electromagnetic fieldshat is equivalent to definition (14).

For the fully coherent electromagnetic fields definedbove, with the insertion of Eqs. (16a) into definition (3)he arbitrary-order correlation tensor in the space-requency domain can be determined as


= �k=1

K

EC�rk,�k� �l=K+1

K+L

HC�rl,�l� �m=K+L+1

K+L+M

EC* �rm,�m�

� �n=K+L+M+1

K+L+M+N

HC* �rn,�n�

��Ex�i��0,�0��K+L�Ex

�i�*��0,�0��M+N�, �18�

hich contains only one term with spatial coordinateseparated and is similar to the case of scalar field (Chap..5.2 of Ref. 5).

. Propagation Theories of Nonstationary Fields Basedn Separated-Coordinate Mode Decomposition inForm of Serieset �s�� s=1,2, . . . denote any complete function groupefined on plane z=0, which can be of real value. Then

�K+L,M+N��e��i� in Eqs. (12) can be decomposed into
1, 2,. . ., K+L+M+N
W 1, 2,. . ., K+L+M+N

�K+L,M+N��e��i� ��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�

= �s1,s2,. . .,sK+L+M+N

c 1, 2,. . ., K+L+M+N

�s1,s2,. . .,sK+L+M+N�

��1,�2, . . . ,�K+L+M+N� �k=1

K+L+M+N

�sk��k�, �19a�

here c 1, 2,. . ., K+L+M+N

�s1,s2,. . .,sK+L+M+N�is of complex value in general.quation (19a) can be rewritten in the tensor form of Eq.

12c) as


= � 1, 2,. . ., K+L+M+Ns1,s2,. . .,sK+L+M+N

�1,�2, . . . ,�K+L+M+Nc 1, 2,. . ., K+L+M+N

�s1,s2,. . .,sK+L+M+N�

��1,�2, . . . ,�K+L+M+N� �k=1

K+L+M+N

�sk��k�. �19b�

quation (19b) is called the SCMD of WJ��K+L,M+N��e��i� in a

orm of series, since each item or mode on the right side ofq. (19b) takes a form with coordinates �i�i=1,2, . . . ,KL+M+N� separated. With Eq. (19a) inserted, Eq. (12a)an become

W�1,�2,. . .,�K+L+M+N


= � 1, 2,. . ., K+L+M+Ns1,s2,. . .,sK+L+M+N

c 1, 2,. . ., K+L+M+N

�s1,s2,. . .,sK+L+M+N��1,�2, . . . ,�K+L+M+N�

� �k=1

K

�P�k k

�e� �rk,�k,�k��sk��k��

l=K+1

K+L

�P�l l

�h� �rl,�l,�l��sl��l��

� �m=K+L+1

K+L+M

�P�m m

�e� �rm,�m,�m��sm��m��*

� �n=K+L+M+1

K+L+M+N

�P�n n

�h� �rn,�n,�n��sn��n��*, �20�

here the linear property of PJ�e��r ,� ,�� and PJ�h��r ,� ,�� issed. To solve W�1,�2,. . .,�K+L+M+N

�K,L,M,N� in Eq. (20), we need onlyo calculate

P� �e��r,�,��s�� = �� · PJ�e��r,�,�� · ��s�� = � · E� �

�s� �r,��,

P� �h��r,�,��s�� = �� · PJ�h��r,�,�� · ��s�� = � · H� �

�s� �r,��,

�21a�

here

E� ��s� �r,�� = PJ�e��r,�,�� · ��s��,

H� ��s� �r,�� = PJ�h��r,�,�� · ��s��. �21b�

y the mathematical equivalence between Maxwell equa-ions (7) and Eqs. (8), Eqs. (21b) are equivalent to

wtpHfiadpfi

EoaB(

wcWardta

WJ

E

W

wut

w


� � E� ��s� �r,�� = i2��r,��H� �

�s� �r,��,

� � H� ��s� �r,�� = − i2��r,��E� �

�s� �r,��,

i.c. E� �i��s��,�� = ��s��, �22�

here E�� i��s�� ,�� represents the transverse component of

he incident or illuminating electric field of each mode onlane z=0. As stated in Subsection 2.C, E� �

�s� �r ,�� and

� ��s� �r ,�� can be treated as fully coherent electromagnetic

elds because they satisfy the Maxwell equations (22),nd can be uniquely determined by solving Eqs. (22) asemonstrated before Eqs. (8), or can be determined by theropagation theories of fully coherent electromagneticelds derived from the Maxwell equations (22).

. Propagation Theories of Nonstationary Fields Basedn Separated-Coordinate Mode Decomposition inForm of Integraly using the Fourier transform, W 1, 2,. . ., K+L+M+N

�K+L,M+N��e��i� in Eqs.
12) can be decomposed into
� �

Bt

wo

W 1, 2,. . ., K+L+M+N

�K+L,M+N��e��i� ��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�

=�� . . .�−�

�

d2f1d2f2 . . . d2fK+L+M+NA 1, 2,. . ., K+L+M+N

��1,�2, . . . ,�K+L+M+N;f1,f2, . . . ,fK+L+M+N�

��k=1

K+L

exp�i2�fk · �k� �m=K+L+1

K+L+M+N

exp�− i2�fm · �m�,

�23a�

here A 1, 2,. . ., K+L+M+Nis of complex value in general and

an be calculated with the inverse Fourier transform of

1, 2,. . ., K+L+M+N

�K+L,M+N��e��i� , fi= fi,xx+ fi,yy�i=1,2, . . . ,K+L+M+N�,nd · means inner product. Equation (23a) can beegarded as an extension of the angular spectrumecomposition of the second-order corrrelations.12 Equa-ion (23a) can be rewritten in the tensor form of Eq. (12c)s

��K+L,M+N��e��i��1,�2, . . . ,�K+L+M+N;�1,�2, . . . ,�K+L+M+N�

= � 1, 2,. . ., K+L+M+N

�� . . .�−�

�

d2f1d2f2 ¯ d2fK+L+M+N�1,�2, . . . ,�K+L+M+N

�A 1, 2,. . ., K+L+M+N��1,�2, . . . ,�K+L+M+N;f1,f2, . . . ,fK+L+M+N��

k=1

K+L


K+L+M+N

exp�− i2�fm · �m�, �23b�

quation (23b) is the SCMD of WJ��K+L,M+N��e��i� in a form of integral. With Eq. (23a) inserted, Eq. (12a) can become

�1,�2,. . .,�K+L+M+N

�K,L,M,N� �r1,r2, . . . ,rK+L+M+N;�1,�2, ¯ ,�K+L+M+N�

= � 1, 2,. . ., K+L+M+N

�� . . .�−�

�

d2f1d2f2 ¯ d2fK+L+M+NA 1, 2,. . ., K+L+M+N��1,�2, . . . ,�K+L+M+N;f1,f2, . . . ,fK+L+M+N�

��k=1

K

�P�k k

�e� �rk,�k,�k�exp�i2�fk · �k�� l=K+1

K+L

�P�l l

�h� �rl,�l,�l�exp�i2�fl · �l�� m=K+L+1

K+L+M

�P�m m

�e� �rm,�m,�m�exp�i2�fm · �m��*

� �n=K+L+M+1

K+L+M+N

�P�n n

�h� �rn,�n,�n�exp�i2�fn · �n��*, �24�

here the linear property of PJ�e��r ,� ,�� and PJ�h��r ,� ,�� issed. To solve W�1,�2,. . .,�K+L+M+N

�K,L,M,N� in Eq. (24), we need onlyo calculate

P� �e��r,�,��exp�i2�f · �� = � · E� ��r,�;f�,

P� �h��r,�,��exp�i2�f · �� = � · H� ��r,�;f�, �25a�

here

E �r,�;f� = PJ�e��r,�,�� · �� exp�i2�f · ��,

H� ��r,�;f� = PJ�h��r,�,�� · �� exp�i2�f · ��. �25b�

y the mathematical equivalence between Maxwell equa-ions (7) and Eqs. (8), Eqs. (25b) are equivalent to

� � E� ��r,�;f� = i2��r,��H� ��r,�;f�,

� � H� ��r,�;f� = − i2��r,��E� ��r,�;f�,

i.c. E�� i� ��,�;f� = � exp�i2�f · ��, �26�

here E�� i� �� ,� ;f� represents the transverse component

f the incident or illuminating electric field of each mode

oMfMwp

FFtfiit

w

W

WJ

w

AttaetVJ

VJ

W

V


n plane z=0. E� ��r ,� ;f� and H� ��r ,� ;f� can be solved byaxwell equations (26), or by the propagation theories of

ully coherent electromagnetic fields derived from theaxwell equations (26). It is notable that the incidentave in Eqs. (26) is a homogenous planar wave of linearolarization.

. Propagation Theories of Stationary Fieldsirst, the propagation problem of arbitrary-order correla-ions of stationary partially coherent electromagneticelds should be presented. We assume that the illuminat-

ng field is stationary, i.e., WJ��K+L,M+N��e��i� in Eqs. (12)

akes the form of


= ��1 + �2 + ¯ + �K+L − �K+L+1 − �K+L+2 − . . .

− �K+L+M+N� � VJ��K+L,M+N��e��i�

��1,�2, . . . ,�K+L+M+N;�2,�3, . . . ,�K+L+M+N�, �27a�

here Eqs. (6) are used, and

ith Eqs. (27) inserted, Eq. (12a) becomes

�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�1,�2, . . . ,�K+L+M+N�

= ��1 + �2 + ¯ + �K+L − �K+L+1 − �K+L+2 − ¯

− �K+L+M+N�VJ�K,L,M,N�

��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�, �28a�

here S

m=K+L+1m m

n=K+L+M+1n

VJ�K,L,M,N��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�

= ��1,�2,. . .,�K+L+M+N

�1,�2, . . . ,�K+L+M+NV�1,�2,. . .,�K+L+M+N

�K,L,M,N�

��r1,r2, . . . ,rK+L+M+N;�2,�3, . . . ,�K+L+M+N�. �28b�

V�1,�2,. . .,�K+L+M+N


= � 1, 2,. . ., K+L+M+N

P�1 1

�e� �r1,�1, �1��k=2

K

P�k k

�e�

��rk,�k,�k� �l=K+1

K+L

P�l l

�h� �rl,�l,�l�

� �m=K+L+1

K+L+M

P�m m

�e�* �rm,�m,�m� �n=K+L+M+1

K+L+M+N

P�n n

�h�* �rn,�n,�n�

� V 1, 2,. . ., K+L+M+N


��1,�2, . . . ,�K+L+M+N;�2,�3, . . . ,�K+L+M+N�, �28c�

�1 = − �2 − �3 − ¯ − �K+L + �K+L+1 + �K+L+2 + ¯ + �K+L+M+N.

�28d�

ccording to Eqs. (6), Eq. (28a) shows that the propaga-ion field is also stationary if the illuminating field is sta-ionary. Through Eqs. (28), the propagation problem ofrbitrary-order correlations of stationary partially coher-nt electromagnetic fields can be expressed as solving theensor VJ�K,L,M,N� in Eq. (28b) with the tensor

��K+L+M+N��e��i� in Eq. (27b) as given.Analogous to SCMD Eq. (23b), VJ�

�K+L,M+N��e��i� has a
CMD in the form of an integral as
��K+L+M+N��e��i��1,�2, . . . ,�K+L+M+N;�2,�3, . . . ,�K+L+M+N�

= � 1, 2,. . ., K+L+M+N

�� ¯�−�

�

d2f1d2f2 . . . d2fK+L+M+N�1,�2, . . . ,�K+L+M+N

�B 1, 2,. . ., K+L+M+N��2,�3, . . . ,�K+L+M+N;f1,f2, . . . ,fK+L+M+N��

k=1

K+L


K+L+M+N

exp�− i2�fm · �m�. �29�

ith SCMD Eq. (29) inserted, Eq. (28c) can become

�1,�2,. . .,�K+L+M+N


= � 1, 2,. . ., K+L+M+N

�� ¯�−�

�

d2f1d2f2 . . . d2fK+L+M+NB 1, 2,. . ., K+L+M+N��2,�3, . . . ,�K+L+M+N;f1,f2, . . . ,fK+L+M+N�

��P�1 1

�e� �r1,�1, �1�exp�i2�f · �1��k=2

K

�P�k k

�e� �rk,�k,�k� exp�i2�fk · �k�� l=K+1

K+L

�P�l l

�h� �rl,�l,�l�exp�i2�fl · �l��

� �K+L+M

�P� �e� �rm,�m,�m�exp�i2�fm · �m��* �

K+L+M+N

�P� �h� �rn,�n,�n�exp�i2�fn · �n�� 30�

n

wcSs

st2d�

3IotS

zdWJ

sd(t=oet

wslicdeate

aHd

Fmedafa


here P� �e��r ,� ,��exp�i2�� ·f� and P�

�h��r ,� ,��exp�i2�� ·f�an be calculated by Eqs. (25) and (26). For the case of theCMD in the form of series, the propagation theories oftationary fields can be obtained analogously.

The propagation theories of stationary fields in thisubsection are almost identical with the propagationheories of nonstationary fields in Subsections 2.D and.E, simply with the correlation tensor W replaced by Vefined in Eqs. (6) and with the frequency �1 replaced by1 defined in Eq. (28d).

. EXAMPLEn this section we provide an example of the propagationf nonstationary partially coherent electromagnetic fieldshrough a dielectric planar interface, using the theories inubsection 2.E.As shown in Fig. 1, the interface is perpendicular to theaxis and is located at z=d, the refractive index of me-

ium I �z�d� is 1, and that of medium II �z�d� is nII.

��K+L,M+N��e��i� of the incident beam on plane z=0 is as-

umed to be known, and WJ��K,L,M,N� of the incident (in me-

ium I), reflected (in medium I) and the transmission fieldin medium II) need to be solved. By the Maxwell equa-ions (26), the transverse component E

�� i� �� ,� ;f�

� exp�i2�f ·�� of the incident electric field of each moden plane z=0 can uniquely determine an incident planarlectromagnetic wave, whose electric and magnetic vec-ors E� �

�i� �r ,� ;f� and H� ��i� �r ,� ;f� are

ig. 1. Propagation of nonstationary partially coherent electro-agnetic fields through a dielectric planar interface at z=d. For

ach mode, E� ��i� �r ,� ;f� ,E� �

�r� �r ,� ;f�, and E� ��t� �r ,� ;f� are the inci-

ent, reflected, and transmission electric fields, respectively; ��i�

nd ��t� are the incident and refractive angle, respectively; and�i� ,f�r�, and f�t� are the spatial frequency of the incident, reflected,nd transmission fields, respectively.

E� ��i� �r,�;f� = E��

�i� �r,�;f� + zEz� ��i� �r,v;f�,

E�� i� �r,�;f� = � exp�i2�f�i� · r�,

Ez� ��i� �r,�;f� = �− x

cos ��i�

cos ��i�− y

cos �i�

cos ��i� � · E�� i� �r,�;f�,

H� ��i� �r,�;f� = ��−1f�i� � E� �

�i� �r,�;f�, �31�

here f�i�=f+zfz= �x cos ��i�+y cos �i�+z cos ��i�� /� is thepatial frequency of the incident wave, �=c /� is the wave-ength in medium I, c is the light velocity in medium I, ��i�

s the incident angle on the interface determined byos2 ��i�+cos2 �i�+cos2 ��i�=1, and � is the magnetic con-uctivity of mediums I and II. By applying the Maxwellquations (26) to the fields on both sides of the interfacend in mediums I and II, the electric and magnetic vec-ors E� �

�r� �r ,� ;f� and H� ��r� �r ,� ;f� of the reflected fields of

ach mode can be determined to be

E� ��r� �r,�;f� = E��

�r� �r,�;f� + zEz� ��r� �r,�;f�,E��

�r� �r,�;f�

= TJ�r� · � exp�i2��−1d cos ��i��exp�i2�f�r� · rd�,

Ez� ��r� �r,�;f� = �x

cos ��i�

cos ��i�+ y

cos �i�

cos ��i� � · E�� r� �r,�;f�,

TJ�r� =1

sin2 ��i��x,y��cos ��i� − cos �i�

cos �i� cos ��i� ��−

tan��i� − ��t��

tan��i� + ��t��0

0 −sin��i� − ��t��

sin��i� + ��t��

�� cos ��i� cos �i�

− cos �i� cos ��i��x

y�,

H� ��r� �r,�;f� = ��−1f�r� � E� �

�r� �r,�;f�, �32�

nd the electric and magnetic vectors E� ��t� �r ,� ;f� and

� ��t� �r ,� ;f� of the transmission fields of each mode can be

etermined to be

E� ��t� �r,�;f� = E��

�t� �r,�;f� + zEz� ��t� �r,�;f�,

E�� t� �r,�;f� = TJ�t� · � exp�i2��−1d cos ��i��exp�i2�f�t� · rd�,

Ez� ��t� �r,�;f� = �− x

cos ��t�

cos ��t�− y

cos �t�

cos ��t� � · E�� t� �r,�;f�,

I−wfw+ta(

chttm

cee

sttn

4Psmmftddepaepta

AI(


TJ�t� =1

sin2 ��i��x,y��cos ��i� − cos �i�

cos �i� cos ��i� ��

sin 2��t�

sin��i� + ��t��cos��i� − ��t��0

02 cos ��i� sin ��t�

sin��i� + ��t��

�� cos ��i� cos �i�

− cos �i� cos ��i��x

y�,

H� ��t� �r,�;f� = ��−1f�t� � E� �

�t� �r,�;f�, �33�

n Eqs. (32) and (33), rd=r−zd, f�r�= �x cos ��i�+y cos �i�

z cos ��i�� /� is the spatial frequency of the reflectedave, f�t�= �x cos ��t�+y cos �t�+z cos ��t�� /�II is the spatial

requency of the transmission wave, �II=� /nII is theavelength in medium II, x cos ��t�+y cos �t�= �x cos ��i�

y cos �i�� /nII, ��t� is the refractive angle in medium II de-ermined by cos2 ��t�+cos2 �t�+cos2 ��t�=1, and TJ�r�and TJ�t�

re determined by the well-known Fresnel formulaChap. 1.5 of Ref. 1).

Until now, the electric and magnetic vectors of the in-ident, reflected, and transmission fields of each modeave been solved by Maxwell equations (26). Thenhrough Eqs. (24) and (25a), the space-frequency correla-ion tensor WJ�K,L,M,N� of the incident, reflected, and trans-ission fields can be determined.It is notable that the theories in this paper are appli-

able to any propagation problem of partially coherentlectromagnetic fields governed by the linear Maxwell
quations (7), including those complex problems without T
olutions with explicit expressions, for example, the elec-romagnetic scattering problem1 and the rigorous diffrac-ion problem of gratings2 under partially coherent illumi-ation.

. CONCLUSIONSropagation theories of arbitrary-order correlations oftationary or nonstationary partially coherent electro-agnetic fields on the basis of the separated-coordinateode decomposition (SCMD) are proposed. SCMD in

orms of series and integral are introduced. Definition ofhe fully coherent electromagnetic field is clarified andiscussed. By using the proposed theories, various well-eveloped theories for the propagation of fully coherentlectromagnetic fields are applicable to the propagation ofartially coherent electromagnetic fields. The theories arepplicable to any propagation problem of partially coher-nt light governed by the linear Maxwell equations. Theropagation of partially coherent electromagnetic fieldshrough a planar dielectric interface is taken as an ex-mple to illustrate the validity of the theories.

PPENDIX An this appendix, we will present the derivations from Eq.11b) to Eqs. (12). The interchange of � and � satisfies

�k=1

K

�l=1

L

akl = ��l1=1

L

a1l1��l2=1

L

a2l2�¯ ��lK=1

L

aKlK�= �

l1,l2. . .,lK=1

L

�k=1

K

aklk. �A1�

hen Eq. (11b) can become

W�1,�2,. . .,�K+L+M+N


= �� 1, 2,. . ., K

�k=1

K

�P�k k

�e� �rk,�k,�k�E k

�i��k,�k�� K+1, K+2,. . ., K+L

�l=K+1

K+L

�P�l l

�h� �rl,�l,�l�E l

�i��l,�l��

K+L+1, K+L+2,. . ., K+L+M

�m=K+L+1

k+L+M

�P�m m

�e�* �rm,�m,�m�E m

�i�*��m,�m��

K+L+M+1, K+L+M+2,. . ., K+L+M+N

�n=K+L+M+1

K+L+M+N

�P�n n

�h�* �rn,�n,�n�E n

�i�*��n,�n��= �

1, 2,. . ., K+L+M+N

��k=1

K

�P�k k

�e� �rk,�k,�k�E k

�i��k,�k�� l=K+1

K+L

�P�l l

�h� �rl,�l,�l�E l

�i��l,�l��

m=K+L+1

K+L+M

�P�m m

�e�* �rm,�m,�m�E m

�i�*��m,�m�� n=K+L+M+1

K+L+M+N

�P�n n

�h�* �rn,�n,�n�E n

�i�*��n,�n��= �

1, 2,. . ., K+L+M+N

��k=1

K

P�k k

�e� �rk,�k,�k�� l=K+1

K+L

P�l l

�h� �rl,�l,�l�� m=K+L+1

K+L+M

P�m m

�e�* �rm,�m,�m�� n=K+L+M+1

K+L+M+N

P�n n

�h�* �rn,�n,�n��

k=1

K

E k

�i��k,�k�� l=K+1

K+L

E l

�i��l,�l�� m=K+L+1

K+L+M

E m

�i�*��m,�m�� n=K+L+M+1

K+L+M+N

E n

�i�*��n,�n��

wfisPJ

pta

Tb

AI(o

w=

Bw

w�

wat


= � 1, 2,. . ., K+L+M+N

��k=1

K

P�k k

�e� �rk,�k,�k�� l=K+1

K+L

P�l l

�h� �rl,�l,�l�� m=K+L+1

K+L+M

P�m m

�e�* �rm,�m,�m�� n=K+L+M+1

K+L+M+N

P�n n

�h�* �rn,�n,�n��

k=1

K+L

E k

�i��k,�k�� m=K+L+1

K+L+M+N

E m

�i�*��m,�m�� , �A2�

E

wvr

F

wvc

W(

E

hich is just Eqs. (12). In Eq. (A2), Eq. (A1) is used for therst equality, the linear property of � � is applied to theecond equality, the linear property of PJ�e��r ,� ,�� and�h��r ,� ,�� is utilized for the third equality, and the linearroperty of PJ�e��r ,� ,��, PJ�h��r ,� ,��, and � � is applied tohe last equality (see Appendix B of Ref. 4). In Eq. (A2), �lso means the composition of operators, namely,

�k=1

K

P�k k

�e� �rk,�k,�k� = P�1 1

�e� �r1,�1,�1�P�2 2

�e� �r2,�2,�2� . . . P�K K

�e�

��rK,�K,�K�. �A3�

he order of the operators on the right side of Eq. (A3) cane changed freely.

PPENDIX Bn this appendix, we will try to derive Eqs. (15) from Eq.14). Equation (14) can be rewritten in a component formf

�Ex�i��1,�1�Ex

�i�*��2,�2�� = Ex,C�i� ��1,�1�Ex,C

�i�*��2,�2�,

�B1a�

�Ey�i��1,�1�Ey

�i�*��2,�2�� = Ey,C�i� ��1,�1�Ey,C

�i�*��2,�2�,

�B1b�

�Ex�i��1,�1�Ey

�i�*��2,�2�� = Ex,C�i� ��1,�1�Ey,C

�i�*��2,�2�,

�B1c�

here E��i�� ,��=xEx

�i�� ,��+yEy�i�� ,�� and E�C

�i� �� ,��xEx,C

�i� �� ,��+yEy,C�i� �� ,��. Equation (B1a) can yield

�Ex�i��1,�1�Ex

�i�*��2,�2�� = �� Ex�i��1,�1� 2�� Ex

�i��2,�2� 2�.

�B2�

y using the Cauchy–Schwarz inequality,15 from Eq. (B2)e can derive

Ex�i��1,�1� = c��1,�2;�1,�2�Ex

�i��2,�2�, �B3�

here c��1 ,�2 ;�1 ,�2� is a complex number dependent on1,�2 ,�1, and �2. Equation (B3) can be rewritten as

Ex�i��,�� = c0��,��Ex

�i��0,�0�, �B4�

here c0�� ,��=c�� ,�0 ;� ,�0�, �0 is a fixed coordinate, �0 isfixed frequency, and �0 and �0 can be selected arbi-

rarily. The insertion of Eq. (B4) into Eq. (B1a) can yield

Ex,C�i� ��1,�1�

c0��1,�1�

Ex,C�i�*��2,�2�

c0*��2,�2�

= � Ex�i��0,�0� 2�. �B5�

quation (B5) is equivalent to

Ex,C�i� ��,��

c0��,��= �� Ex

�i��0,�0� 2�exp�− i�x�, �B6�

here �x is independent of � and �, and can be of any realalue. The insertion of Eq. (B6) into Eq. (B4) shows theelation between Ex

�i�� ,�� and Ex,C�i� �� ,�� as

Ex�i��,�� = Ex,C

�i� ��,��Ex�i��0,�0�,

Ex�i��0,�0� =

Ex�i��0,�0�

�� Ex�i��0,�0� 2�

exp�i�x�. �B7�

rom Eq. (B1b), it can be similarly derived that

Ey�i��,�� = Ey,C

�i� ��,��Ey�i��0,�0�,

Ey�i��0,�0� =

Ey�i��0,�0�

�� Ey�i��0,�0� 2�

exp�i�y�, �B8�

here �y is independent of � and �, and can be of any realalue. The insertion of Eqs. (B7) and (B8) into Eq. (B1c)an yield

�Ex�i��0,�0�Ey

�i�*��0,�0�� = 1. �B9�

ith the definitions of Ex�i��0 ,�0� and Ey

�i��0 ,�0� in Eqs.B7) and (B8) inserted, Eq. (B9) can become

exp�i��x − �y��Ex

�i��0,�0�Ey�i�*��0,�0��

�� Ex�i��0,�0� 2�� Ey

�i��0,�0� 2�= 1.

�B10�

quation (B10) shows

By

wi

WE

So

w

ATdRfood

ef

R

1

1

1

1

1

1


�Ex�i��0,�0�Ey

�i�*��0,�0�� = �� Ex�i��0,�0� 2�� Ey

�i��0,�0� 2�.

�B11�

y using the Cauchy–Schwarz inequality,15 Eq. (B11) canield

Ey�i��0,�0� = cxyEx

�i��0,�0�, �B12�

here cxy is a complex number. With Eq. (B12) insertednto Eq. (B10), we can obtain

exp�i��x − �y��cxy

*

cxy = 1. �B13�

ith Eqs. (B12) and (B13) inserted into the expression of˜

y�i��0 ,�0� in Eqs. (B8), we have

Ey�i��0,�0� = Ex

�i��0,�0�. �B14�

o Eqs. (B7) and (B8) can be rewritten in the vector formf

E��i��,�� = E�C

�i� ��,��Ex�i��0,�0�, �B15�

hich is just Eqs. (15).

CKNOWLEDGMENTShis research is supported by the Natural Science Foun-ation of Tianjin (grant 06YFJMJC01500), by the Openesearch Fund of Key Laboratory of Opto-electronic In-

ormation Science and Technology of Education Ministryf China (grant 2005-04), and by the Fund for the Devel-pment Project of Science and Technology of Tianjin (un-er grant 043103011).

Corresponding author Haitao Liu can be reached by-mail at [email protected]; phone, 8622-23506422;
ax, 8622-23502275.
EFERENCES1. M. Born and E. Wolf, Principles of Optics, 5th ed.

(Pergamon, l975).2. L. Li, “Use of Fourier series in the analysis of

discontinuous periodic structures,” J. Opt. Soc. Am. A 13,1870–1876 (1996).

3. A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonalmethod for calculation of coherence functions,” Phys. Rev.Lett. 95, 043904 (2005).

4. H. Liu, G. Mu, and L. Lin, “Propagation theories ofpartially coherent electromagnetic fields based on coherentor separated-coordinate mode decomposition,” J. Opt. Soc.Am. A 23, 2208–2218 (2006).

5. L. Mandel and E. Wolf, Optical Coherence and QuantumOptics (Cambridge U. Press, 1995).

6. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi,and G. Guattari, “Coherent-mode decomposition ofpartially polarized, partially coherent sources,” J. Opt. Soc.Am. A 20, 78–84 (2003).

7. J. Tervo, T. Setala, and A. T. Friberg,“Theory of partiallycoherent electromagnetic fields in the space-frequencydomain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).

8. T. Setala, J. Lindberg, K. Blomstedt, J. Tervo, and A. T.Friberg, “Coherent-mode representation of a statisticallyhomogeneous and isotropic electromagnetic field inspherical volume,” Phys. Rev. E 71, 036618 (2005).

9. J. W. Goodman, Statistical Optics (Wiley, 1985).0. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energy

spectrum of a nonstationary ensemble of pulses,” Opt. Lett.29, 394–396 (2004).

1. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatiallyand spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005).

2. J. Tervo and J. Turunen, “Angular spectrum representationof partially coherent electromagnetic fields,” Opt. Commun.209, 7–16 (2002).

3. R. K. Luneberg, Mathematical Theory of Optics (Universityof California Press, 1964), pp. 319–320.

4. O. Korotkova and E. Wolf, “Spectral degree of coherence ofa random three-dimensional electromagnetic field,” J. Opt.Soc. Am. A 21, 2382–2385 (2004).

5. T. S. Blyth and E. F. Robertson, Further Linear Algebra(Springer, 2002), Chap. 1.

propagation theories of arbitrary-order correlations of partially coherent electromagnetic fields...

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