propagation theories of partially coherent electromagnetic fields based on coherent or...
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2208 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
Propagation theories of partially coherentelectromagnetic fields based on coherent or
separated-coordinate mode 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 January 25, 2006; accepted March 31, 2006; posted April 26, 2006 (Doc. ID 67418)
Propagation theories of partially coherent electromagnetic fields based on coherent mode decomposition orseparated-coordinate mode decomposition are proposed. With the proposed propagation theories, various pow-erful theories for the propagation of fully coherent electromagnetic fields can be used for the propagation ofpartially coherent electromagnetic fields. The proposed theories are applicable to any propagation problem ofpartially coherent electromagnetic fields governed by linear Maxwell equations. Some examples are providedto illustrate the validity of the proposed theories. © 2006 Optical Society of America
OCIS codes: 030.6600, 030.4070, 260.2110.
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. INTRODUCTIONhe propagation problem of light plays a significant role
n optics.1–3 Compared with various powerful theories forhe propagation of fully coherent light, such as the geo-etrical optics (Chap. 3 of Ref. 1) and the rigorous diffrac-
ion theories (Chap. 11 of Ref. 1 and Ref. 4), theories forhe propagation of partially coherent light seem to beore deficient. Coherent mode decomposition (CMD)2
ight be a solution, which shows that partially coherentight can be treated as a superposition of a series of fullyoherent light modes, so that the various powerful theo-ies for the propagation of fully coherent light might bepplicable to the propagation of partially coherent light.MD of scalar fields has been applied to the propagationf partially coherent scalar waves.5 CMD of partially co-erent electromagnetic fields has been reported in recentears.6–8 There is also some work about the propagationf partially coherent electromagnetic fields.9,10 However,ropagation theories of partially coherent electromag-etic fields based on CMD have not been established yet.In this paper, propagation theories of partially coherent
lectromagnetic fields based on CMD are proposed. Givenhat CMD requires solution of the eigenvalue integralquations, propagation theories of partially coherent elec-romagnetic fields based on a separated-coordinate modeecomposition (SCMD) are also proposed. SCMD avoidshe need to solve the eigenvalue integral equations and is
generalization of CMD. With the proposed theoriesased on CMD or SCMD, various propagation theories ofully coherent electromagnetic fields can be applicable tohe propagation of partially coherent electromagneticelds. The proposed theories are applicable to any propa-ation problem of partially coherent electromagneticelds governed by linear Maxwell equations.
1084-7529/06/092208-11/$15.00 © 2
In Section 2, propagation theories based on the twoinds of mode decomposition are presented. In Section 3,wo examples are provided. In Section 4, conclusions areummarized.
. PROPAGATION THEORIES. Presentation of the Propagation Problem
n this paper, the random electric vector and the ran-om magnetic vector of the partially coherent electro-agnetic fields in the space-frequency domain are
enoted by E�r ,��= „Ex�r ,�� ,Ey�r ,�� ,Ez�r ,��…T and�r ,��= „Hx�r ,�� ,Hy�r ,�� ,Hz�r ,��…T, respectively, wheremeans transpose of a matrix, r= �x ,y ,z�T is the spatial
oordinate, and � is the frequency. The electric, magnetic,nd mixed cross-spectral density tensors are defined as2
WJ�e��r1,r2,�� = �E�r1,��E+�r2,��� = ��Ep�r1,��Eq*�r2,����,
�1a�
WJ�h��r1,r2,�� = �H�r1,��H+�r2,��� = ��Hp�r1,��Hq*�r2,����,
�1b�
WJ�m��r1,r2,�� = �E�r1,��H+�r2,���
= ��Ep�r1,��Hq*�r2,����, p,q � �x,y,z�,
�1c�
here � � means the ensemble average, + means transposend conjugate of a matrix, * means complex conjugate,nd � � denotes a matrix. A transverse electric cross-pectral density tensor WJ�e��i��� ,� ,�� of the incident or
� 1 2006 Optical Society of America
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Liu et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2209
lluminating electromagnetic fields on a transverse plane=0 is defined as
WJ��e��i���1,�2,�� = �E�
�i���1,��„E��i���2,��…+� = ��Ep
�i���1,��„Eq�i�
���2,��…*��, p,q � �x,y�, �2�
here E��i��� ,��= „Ex
�i��� ,�� ,Ey�i��� ,��…T is the transverse
omponent of the incident random electric field on plane=0 and �= �x ,y ,0�T is the spatial coordinate on plane z0. It will be shown by the propagation theories in Sec-
ion 3 that WJ��e��i���1 ,�2 ,�� can uniquely determine
�e��r1 ,r2 ,��, WJ�h��r1 ,r2 ,��, and WJ�m��r1 ,r2 ,�� if the me-ium geometry is assumed to be known. Thus, for theropagation problem of partially coherent electromag-etic fields, WJ�
�e��i���1 ,�2 ,��on a transverse plane is usu-lly assumed to be known and WJ�e��r1 ,r2 ,��,
�h��r1 ,r2 ,��, and WJ�m��r1 ,r2 ,�� need to be solved.11
The Maxwell equations governing the linear propaga-ion of the random electromagnetic fields take a linearorm of1
� � E�r,�� = i2���H�r,��,
� � H�r,�� = − i2���E�r,��, b . c . E��i���,��, �3�
here �= �� /�x ,� /�y ,� /�z�T, � and � are the magnetic con-uctivity and the permittivity of the medium, respec-ively, and “b.c. E�
�i��� ,��” means that the transverse com-onent E�
�i��� ,�� of the incident or illuminating randomlectric field on plane z=0 acts as the boundary conditionf illumination. Because E�
�i��� ,�� can uniquely determineoth the electric and the magnetic vectors of the incidentr illuminating field11 and the medium geometry of � andis assumed to be known, Eqs. (3) should have a unique
olution dependent on E�i��� ,�� by physical intuition.
�2� �y � z
ased on this point, the Maxwell equations (3) can be re-ritten with mathematical equivalence in a solution formf
E�r,�� = PJ�e��r,�,��E��i���,��, �4a�
H�r,�� = PJ�h��r,�,��E��i���,��. �4b�
he abstract operators PJ�e��r ,� ,�� and PJ�h��r ,� ,�� canepresent any linear propagation physics governed by theinear Maxwell equations (3). By using the linear prop-rty of the Maxwell equations (3), it can be proved in Ap-endix A that PJ�e��r ,� ,�� and PJ�h��r ,� ,�� are both 3�2ensor operators, with all of their six elements being lin-ar operators. For example, in the homogeneous half-pace z�0, E�r ,�� and H�r ,�� can be uniquely deter-ined by E�
�i��� ,�� as12,13
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,�,��
�yEy
�i���,���d2�,
H�r,�� = �i2����−1 � � E�r,��, �5�
here G�r ,� ,��=exp�ik �r−� � � / �r−�� with k=2�� /c.quations (5) are equivalent to the Maxwell equations (3)
n the homogeneous half-space z�0 and can be expressedn the form of Eqs. (4) with
PJ�e��r,�,�� = −
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
� , �6a�
PJ�h��r,�,�� =1
i2��� 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 d2��2G�r,�,��
−1 d2�
�2G�r,�,�� � . �6b�
2� �x � z
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2210 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
. Propagation Theories Based on Coherent Modeecomposition
t is straightforward that
„WJ��e��i���2,�1,��…+ = WJ�
�e��i���1,�2,��, �7a�
f+��1�WJ��e��i���1,�2,��f��2�d2�1d2�2
=� f+��1�E��i���1,��
�„E��i���2,��…+f��2�d2�1d2�2�
=�� f+���E��i���,��d2���2
� 0
" f��� = „fx���,fy���…, �7b�
here " means “for any”. Relations (7) show that
��e��i���1 ,�2 ,�� corresponds to a Hermitian, positive defi-
ite or positive semidefinite operator in the Hilbert space,o WJ�
�e��i���1 ,�2 ,�� has a CMD of6,7
WJ��e��i���1,�2,�� = �
n�n���En�
�i� ��1,��„En��i� ��2,��…+, �8�
here �n����0 are the eigenvalues and En��i� �� ,�� are the
orresponding complete and orthogonal eigenfunctions of
ectorial value, which can be determined by solving the tigenvalue integral equations of
WJ��e��i���1,�2,��En�
�i� ��2,��d2�2 = �n���En��i� ��1,��,
„Em��i� ��,��…+En�
�i� ��,��d2� = mn, �9�
here mn is the Kronecker delta symbol. Each item ofhe series in CMD (8) satisfies relations (7) and thus is ahysical mode. Equations (9) are only sufficient but notecessary conditions of Eq. (8). For example, Eq. (8) cane rewritten as
WJ��e��i���1,�2,�� = �
n�n���En�
�i� �1,��„En��i� �2,��…+, �10�
here �n���=cn�n���, En��i� �� ,��=En�
�i� �� ,�� /�cn, and cn cane any positive constant. It is obvious that �n��� and
˜n��i� �� ,�� in CMD (10) may not satisfy Eqs. (9) with �n���nd En�
�i� �� ,�� replaced by �n��� and En��i� �� ,��, respec-
ively.Based on CMD (8) and the propagation equations (4),
e will try to derive the equations governing the propa-ation of partially coherent electromagnetic fields in theollowing. Let WJ�e��r1 ,r2 ,��= �Wpq
�e��r1 ,r2 ,���, E�r ,���Ep�r ,���, E�
�i��� ,��= �Ek�i��� ,���, PJ�e��r ,� ,��= �Ppk
�e��r ,� ,���,
��e��i���1 ,�2 ,��= �Wkl
�e��i���1 ,�2 ,���, and En��i� �� ,��
�Enk�i� �� ,���, where p ,q� �x ,y ,z�and k , l� �x ,y�. The inser-
ion of Eq. (4a) into definition (1a) yields that
Wpq�e��r1,r2,��=
1
�Ep�r1,��Eq*�r2,���=
2��k
Ppk�e��r1,�1,��Ek
�i���1,����i
Pql�e��r2,�2,��El
��i���2,���*�=3��
kPpk
�e��r1,�1,��Ek�i���1,����
l„Pql
�e��r2,�2,��…*„El
�i���2,��…*��=4��
k�
lPpk
�e��r1,�1,��„Pql�e��r2,�2,��…*Ek
�i���1,��„El�i���2,��…*�
=5
�k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*�Ek�i���1,��„El
�i���2,��…*�
=6
�k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*Wkl�e��i���1,�2,��
=7
�k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*�n
�n���Enk�i� ��1,��„Enl
�i���2,��…*�=8
�n
�n����k
Ppk�e��r1,�1,��Enk
�i� ��1,����l„Pql
�e��r2,�2,��…*„Enl
�i���2,��…*�=9
�n
�n����k
Ppk�e��r1,�1,��Enk
�i� ��1,����l
Pql�e��r2,�2,��Enl
�i���2,���*=10
�n
�n���Enp�r1,��Enq* �r2,��, �11a�
here Enp�r ,��=�k
Ppk�e��r ,� ,��Enk
�i� �� ,��. The derivations in Eq. (11a) seem to be a little tedious, but they can be rewritten in
he more concise but more formal manner of
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Liu et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2211
WJ�e��r1,r2,��=1
�E�r1,��„E�r2,��…+�=2
��PJ�e��r1,�1,��E��i���1,����PJ�e��r2,�2,��E�
�i���2,���+�
=3,4
�PJ�e��r1,�1,���E��i���1,��„E�
�i���2,��…+�„PJ�e��r2,�2,��…+�
=5
PJ�e��r1,�1,���E��i���1,��„E�
�i���2,��…+�„PJ�e��r2,�2,��…+
=6
PJ�e��r1,�1,��WJ��e��i���1,�2,��„PJ�e��r2,�2,��…+
=7
PJ�e��r1,�1,���n
�n���En��i� ��1,��„En�
�i� ��2,��…+�„PJ�e��r2,�2,��…+
=8,9
�n
�n����PJ�e��r1,�1,��En��i� ��1,����PJ�e��r2,�2,��En�
�i� ��2,���+=10
�n
�n���En�r1,��En+�r2,��, �11b�
he
CMTgdtp
wccEpcrei(Ct
tfi
w
here
En�r,�� = PJ�e��r,�,��En��i� ��,��. �12�
he numbers 1–10 above the equal signs in Eqs. (11) areor easy correspondence between Eqs. (11a) and (11b). Inqs. (11), the linear property of the tensor operator PJ�e�
�r ,� ,�� is used, definition (2) is applied to equality 6,nd CMD (8) is used for equality 7. Equality 5 in Eqs. (11)s not obvious, which can be proved in Appendix B basedn the linear property of the tensor operator PJ�e��r ,� ,��.he derivations in Eqs. (11) can be easily comprehendedith Eq. (6a) as an example of PJ�e��r ,� ,��. Analogous toqs. (11) and (12), WJ�h��r1 ,r2 ,�� and WJ�m��r1 ,r2 ,�� can be
alculated by
WJ�h��r1,r2,�� = �n
�n���Hn�r1,��Hn+�r2,��,
WJ�m��r1,r2,�� = �n
�n���En�r1,��Hn+�r2,��, �13�
here
Hn�r,�� = PJ�h��r,�,��En��i� ��,��. �14�
quations (11)–(14) show that WJ�e��r1 ,r2 ,��, WJ�h�
�r1 ,r2 ,��, and WJ�m��r1 ,r2 ,�� can be uniquely deter-ined by WJ�
�e��i���1 ,�2 ,�� if the medium geometry is as-umed to be known, as mentioned following Eq. (2). Iniew of the mathematical equivalence between the Max-ell equations (3) and the propagation equations (4), Eqs.
12) and (14) are equivalent to
� � En�r,�� = i2���Hn�r,��,
� � Hn�r,�� = − i2���En�r,��, b.c.En��i� ��,��. �15�
quations (11b), (13), and (15) are the main conclusions ofhis subsection. The quantities En�r ,�� and Hn�r ,�� cane treated as fully coherent electromagnetic fields be-ause they satisfy the Maxwell equations (15), and theyan be uniquely determined by solving the Maxwell equa-ions (15) as demonstrated following Eqs. (3) or can be de-ermined by the various propagation theories of fully co-
erent electromagnetic fields derived from the Maxwellquations (15).
. Propagation Theories Based on Separated-Coordinateode Decompositionhe theories in Subsection 2.B require one to solve the ei-envalue equations (9), which is sometimes difficult. Theerivations in Subsection 2.B imply that a more generalype of mode decomposition may also be applicable to theropagation problem, which is
WJ��e��i���1,�2,�� = �
ncn���En�
����i���1,��„En�����i��2,��…+,
�16�
here En�����i��� ,�� and En�
����i��� ,�� can be two differentomplex vectors with x and y components and cn��� areomplex numbers dependent on frequency � in general.quation (16) is called separated-coordinate mode decom-osition (SCMD) in this paper, since each item of the de-omposition takes a form with coordinates �1 and �2 sepa-ated. Equations (16) is a generalization of CMD (8). Notvery mode of decomposition (16) may satisfy the Hermit-an and positive definite or positive semidefinite relations7) and thus be a physical mode. It is shown in Appendix
that SCMD avoids the need to solve the eigenvalue in-egral equations and is much easier to obtain than CMD.
In the following, we will derive the equations governinghe propagation of partially coherent electromagneticelds based on SCMD. Analogous to Eqs. (11), we have
WJ�e��r1,r2,�� = PJ�e��r1,�1,��WJ��e��i���1,�2,��„PJ�e��r2,�2,��…+
= PJ�e��r1,�1,���n
cn���En�����i���1,��
�„En�����i���2,��…+�„PJ�e��r2,�2,��…+
= �n
cn����PJ�e��r1,�1,��En�����i���1,���
��PJ�e��r2,�2,��En�����i���2,���+
= �n
cn���En����r1,��„En
����r2,��…+, �17�
here
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2212 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
En����r,�� = PJ�e��r,�,��En�
����i���,��,
En����r,�� = PJ�e��r,�,��En�
����i���,��. �18�
n Eq. (17), the linear property of the tensor operator�e��r ,� ,�� is used, and SCMD (16) is used for the second
quality. Analogous to Eqs. (17) and (18), WJ�h��r1 ,r2 ,��nd WJ�m��r1 ,r2 ,�� can be calculated by
WJ�h��r1,r2,�� = �n
cn���Hn����r1,��„Hn
����r2,��…+,
WJ�m��r1,r2,�� = �n
cn���En����r1,��„Hn
����r2,��…+, �19�
here
Hn����r,�� = PJ�h��r,�,��En�
����i���,��,
Hn����r,�� = PJ�h��r,�,��En�
����i���,��. �20�
quations (17)–(20) show that WJ�e��r1 ,r2 ,��,�h��r1 ,r2 ,��, and WJ�m��r1 ,r2 ,�� can be uniquely deter-ined by WJ�
�e��i���1 ,�2 ,�� if the medium geometry is as-umed to be known, as mentioned following Eq. (2) or14). In view of the mathematical equivalence betweenhe Maxwell equations (3) and the propagation equations4), Eqs. (18) and (20) are equivalent to
� � En����r,�� = i2���Hn
����r,��,
� � Hn����r,�� = − i2���En
����r,��,
b.c. En�����i���,��, �21a�
� � En����r,�� = i2���Hn
����r,��,
� � Hn����r,�� = − i2���En
����r,��,
b.c. En�����i���,��. �21b�
quations (17), (19), and (21) are the main conclusionsf this subsection. The quantities En
����r ,��, Hn����r ,��,
n����r ,��, and Hn
����r ,�� can be treated as fully coherentlectromagnetic fields because they satisfy the Maxwellquations (21), and they can be uniquely determined byolving the Maxwell equations (21) as demonstrated fol-owing Eqs. (3) or can be determined by the various theo-ies for the propagation of fully coherent electromagneticelds derived from the Maxwell equations (21).
. EXAMPLES. Propagation of Random Planar Electromagneticields through a Planar Interfaces shown in Fig. 1, let us consider the propagation of aandom planar electromagnetic field from medium I toedium II with planar interface, using the theories inubsection 2.B based on CMD. The interface is perpen-icular to the z axis and located at z=d, the refractive in-ex of medium I is assumed to be 1, and the refractive in-ex of medium II is assumed to be n .
IIThe electric field of the incident random planar electro-agnetic field on plane z=0 is
E�i���,�� = E0�i� exp�i2�f�i�T��, �22�
here �= �x ,y ,0�T is the coordinate on plane z=0,�i��� ,��= �Ex
�i��� ,�� ,Ey�i��� ,�� ,Ez
�i��� ,���T, E0�i�= �E0,x
�i� ,E0,y�i� ,
0,z�i� �T is a random vector, and f�i�= �fx
�i� , fy�i� ,0�T is the trans-
erse spatial frequency. Then the transverse electricross-spectral density tensor of the incident beam onlane z=0 is
WJ��e��i���1,�2,�� = ��Ep
�i���1,��„Eq�i���2,��…*��
= AJ exp�i2�f�i�T��1 − �2��, �23�
here p ,q� �x ,y� and AJ= ��E0,p�i� E0,q
�i�*�� is a constant tensor.or the propagation problem, WJ�
�e��i���1 ,�2 ,�� is assumedo be known, and the cross-spectral density tensors of thencident, reflected, and transmission waves in media Ind II need to be solved.The CMD of WJ�
�e��i���1 ,�2 ,�� of this example can be ob-ained in a simple way, similar to solving eigenvalue inte-ral equations (9). In view of relations (7), AJ should be aermitian and positive definite or positive semidefinite
ensor. So AJ can be decomposed into
AJ = �1a1a1+ + �2a2a2
+, �24�
here an and �n (n=1,2 and �n�0) are the eigenvectorsnd the eigenvalues of AJ, which can be determined byolving AJan=�nan and am
+ an=mn. Then the CMD of
��e��i���1 ,�2 ,�� is
WJ��e��i���1,�2,�� = �1E1�
�i� ��1,��„E1��i� ��2,��…+
+ �2E2��i� ��1,��„E2�
�i� ��2,��…+,
En��i� ��,�� = an exp�i2�f�i�T��, n = 1,2. �25�
y the Maxwell equations (15), the two coherent modes of
��e��i���1 ,�2 ,�� can uniquely determine two incident fully
oherent planar electromagnetic fields with the same spa-ial frequency of f�i�= �f�i� , f�i� ,�1/�2− f�i�2− f�i�2�T, where �
Fig. 1. Scheme of the example in Subsection 3.A.
x y x y
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=llzi(�
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Liu et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2213
c /� is the wavelength in medium I and c is the light ve-ocity in medium I. As shown in Fig. 1, the x axis is se-ected within the incident plane that contains f�i� and the
axis. Thus f�i�=�−1�sin ��i� ,0 ,cos ��i��T, where ��i� is thencident angle on the interface. By the Maxwell equations15), the electric and magnetic vectors En
�i��r ,�� and Hn�i�
�r ,�� of the incident field of each mode is
En�i��r,�� = �„En�
�i� �r,��…T, En,z�i� �r,���T,
En��i� �r,�� = an exp�i2�f�i�Tr�,
En,z�i� �r,�� = �− tan ��i�, 0� · En�
�i� �r,��,
Hn�i��r,�� = ����−1f�i� � En
�i��r,��, n = 1,2, �26�
here r= �x ,y ,z�T, En��i� �r ,�� and En,z
�i� �r ,�� are the trans-erse and z components of En
�i��r ,��, respectively, and � ishe magnetic conductivity of media I and II. By applyinghe Maxwell equations (15) to the fields on both sides ofhe interface and in media I and II, the electric and mag-etic vectors En
�r��r ,�� and Hn�r��r ,�� of the reflected field of
ach mode can be determined as
En�r��r,�� = �„En�
�r� �r,��…T, En,z�r� �r,���T,
En��r� �r,�� = TJ�r�an exp�i2��−1d cos ��i��exp�i2�f�r�Trd�,
En,z�r� �r,�� = �tan ��i�, 0� · En�
�r� �r,��,
TJ�r� = −tan���i� − ��t��
tan���i� + ��t��0
0 −sin���i� − ��t��
sin���i� + ��t��� ,
Hn�r��r,�� = ����−1f�r� � En
�r��r,��, n = 1,2, �27�
nd the electric and magnetic vectors En�t��r ,�� and
n�t��r ,�� of the transmission field of each mode can be de-
ermined as
En�t��r,�� = �„En�
�t� �r,��…T, En,z�t� �r,���T,
En��t� �r,�� = TJ�t�an exp�i2��−1d cos ��i��exp�i2�f�t�Trd�,
En,z�t� �r,�� = �− tan ��t�, 0� · En�
�t� �r,��,
TJ�t� = sin 2��t�
sin���i� + ��t��cos���i� − ��t��0
02 cos ��i� sin ��t�
sin���i� + ��t��� ,
Hn�t��r,�� = ����−1f�t� � En
�t��r,��, n = 1,2. �28�
n Eqs. (27) and (28), En�� � �r ,�� and En,z
� ��r ,�� are theransverse and z components of E� ��r ,�� � =r , t�, respec-
nively, rd= �x ,y ,z−d�T, f�r�=�−1�sin ��i� ,0 ,−cos ��i��T is thepatial frequency of the reflected fields, f�t�
�II−1�sin ��t� ,0 ,cos ��i��T is the spatial frequency of the
ransmission fields, ��t�=arcsin�nII−1 sin ��i�� is the refrac-
ive angle, �II=� /nII is the wavelength in medium II, and�r�and TJ�t� are determined by the well-known Fresnel for-ula (Chap. 1.5 of Ref. 1). By Eqs. (11b), (13), and (15),
he electric, magnetic, and mixed cross-spectral densityensors of the incident � = i�, reflected � =r�, and trans-ission � = t� waves are
WJ�e�� ��r1,r2,�� = �1E1� ��r1,��„E1
� ��r2,��…+
+ �2E2� ��r1,��„E2
� ��r2,��…+,
WJ�h�� ��r1,r2,�� = �1H1� ��r1,��„H1
� ��r2,��…+
+ �2H2� ��r1,��„H2
� ��r2,��…+,
WJ�m�� ��r1,r2,�� = �1E1� ��r1,��„H1
� ��r2,��…+
+ �2E2� ��r1,��„H2
� ��r2,��…+. �29�
o provide a numerical example, we select AJ= �2 11 2 �
nd nII=1.52 (nII is identical with that of Fig. 1.12 inhap. 1.5.3 of Ref. 1 for comparison) and calculate the re-ectance R and the transmittance T. R and T are defineds (Chap. 1.5.3 of Ref. 1)
R =Tr WJ�e��r��r�,r�,��
Tr WJ�e��i��r,r,��, T =
nII cos ��t�
cos ��i�
Tr WJ�e��t��r�,r�,��
Tr WJ�e��i��r,r,��.
�30�
he numerical results show that the energy conservationelation of R+T=1 is satisfied for any ��i�� �0° ,90° �. Ashown by the solid curve in Fig. 2, R of the example is be-ween that of a TE wave with electric vector perpendicu-ar to the incident plane and that of a TM wave with elec-ric vector parallel to the incident plane. It is notable thatith ��i� approaching 90°, R of the example approaches
hat of a TM wave, which can be explained in the follow-ng. The random TM component E0,TM
�i� and the randomE component E0,TE
�i� of E0�i� in Eq. (22) can be expressed as
ig. 2. Reflectance R of the numerical example in Subsection.A.
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2214 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
0,TM�i� =E0,x
�i� / cos ��i� and E0,TE�i� =E0,y
�i� . Then we have
��E0,TM�i� �2� = ��E0,x
�i� �2�/cos2 ��i� = 2/cos2 ��i�,
��E0,TE�i� �2� = ��E0,y
�i� �2� = 2. �31�
f ��i� approaches 90°, we have ��E0,TM�i� �2�� ��E0,TE
�i� �2�, or thentensity of the TM component is far larger than that ofhe TE component for the incident random field, and thus
of the example approaches that of the TM wave.
. Propagation of Arbitrary Partially Coherentlectromagnetic Fields through a Planar Interfaceor the example in this subsection, the medium geometry
s the same as that in Subsection 3.A, as shown in Fig. 1.he interface is perpendicular to the z axis and located at=d, the refractive index of medium I is assumed to be 1,nd the refractive index of medium II is assumed to beII.The incident beam on plane z=0 is assumed to be an
rbitrary partially coherent electromagnetic field, whichs described by the transverse electric cross-spectral den-ity tensor WJ�
�e��i���1 ,�2 ,��. Here �= �x ,y ,0�T is the coordi-ate on plane z=0. For the propagation problem,
��e��i���1 ,�2 ,�� is assumed to be known, and the cross-
pectral density tensors of the incident, reflected, andransmission waves in media I and II need to be solved. Inhe following, we will try to provide a solution of the prob-em with explicit expressions by using the propagationheories based on SCMD. However, it seems to be difficulto obtain a solution of the problem with explicit expres-ions if we use the propagation theories based on CMD.
As shown in Appendix C, the SCMD of WJ��e��i���1 ,�2 ,��
n the form of an integral is
WJ��e��i���1,�2,�� = E1�
����i���1,�;f1,f2�
�„E1�����i���2,�;f2�…+d2f1d2f2
+ E2�����i���1,�;f1,f2�
�„E2�����i���2,�;f2�…+d2f1d2f2, �32a�
here
E1�����i���1,�;f1,f2� = Axx
�e��f1,f2,��
Ayx�e��f1,f2,���exp�i2�f1
T�1�,
E1�����i���2,�;f2� = 1
0�exp�i2�f2T�2�,
E2�����i���1,�;f1,f2� = Axy
�e��f1,f2,��
Ayy�e��f1,f2,��� exp�i2�f1
T�1�,
E2�����i���2,�;f2� = 0
1�exp�i2�f2T�2�. �32b�
he quantities in Eqs. (32) are explained in Appendix C.s demonstrated in Appendix C, SCMD (32) comes from
he well-known angular spectrum decomposition of theeld, i.e., all the modes of the decomposition are planaraves in the angular spectrum (Chap. 3.2 of Ref. 2).quations (32b) have a unified expression of
En����i���,�� = En�
���i� exp�i2�f�i�T��, ٠� ��, � �, n � �1,2�,
�33�
here f�i�= �fx�i� , fy
�i� ,0�T is the transverse spatial frequencynd En�
���i� is independent of �. By the Maxwell equations21), Eq. (33) can determine an incident planar electro-agnetic wave, whose electric and magnetic vectors
n���i��r ,�� and Hn
���i��r ,�� are
En�Ù��i��r,�� = �„En�
�Ù��i��r,��…T, En,z�Ù��i��r,���T,
En����i��r,�� = En�
���i� exp�i2�f�i�Tr�,
En,z�Ù��i��r,�� = −
cos �i�
cos ��i�, −
cos ��i�
cos ��i� � · En��Ù��i��r,��,
Hn�Ù��i��r,�� = ����−1f�i� � En
���i��r,��,
٠� ��, � �, n � �1,2�, �34�
here r= �x ,y ,z�T, En����i��r ,�� and En,z
���i��r ,�� are theransverse and z components of En
���i��r ,��, respectively,�i�= �fx
�i� , fy�i� , fz
�i��T=�−1�cos �i� , cos ��i� , cos ��i��T is the spa-ial 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 media I and II. By applying the Maxwellquations (21) to the fields on both sides of the interfacend in media I and II, the electric and magnetic vectors
n���r��r ,�� and Hn
���r��r ,�� of the reflected fields of eachode can be determined to be
En�Ù��r��r,�� = �„En�
�Ù��r��r,��…T, En,z�Ù��r��r,���T
En����r��r,�� = TJ�r�En�
�Ù��i� exp�i2��−1d cos ��i��exp�i2�f�r�Trd�,
En,z���r��r,�� = cos �i�
cos ��i�,
cos ��i�
cos ��i� � · En��Ù��r��r,��,
TJ�r� =1
sin2 ��i�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�� ,
![Page 8: Propagation theories of partially coherent electromagnetic fields based on coherent or separated-coordinate mode decomposition](https://reader031.vdocuments.mx/reader031/viewer/2022020407/5750821f1a28abf34f96c26c/html5/thumbnails/8.jpg)
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Liu et al. Vol. 23, No. 9 /September 2006 /J. Opt. Soc. Am. A 2215
Hn�Ù��r��r,�� = ����−1f�r� � En
���r��r,��,
٠� ��, � �, n � �1,2�, �35�
nd the electric and magnetic vectors En���t��r ,�� and
n���r��r ,�� of the transmission fields of each mode can be
etermined to be
En�Ù��t��r,�� = �„En�
�Ù��t��r,��…T, En,z�Ù��t��r,���T,
En����t��r,�� = TJ�t�En�
�Ù��i� exp�i2��−1d cos ��i��exp�i2�f�t�Trd�,
En,z���t��r,�� = cos �t�
cos ��t�, −
cos ��t�
cos ��t� � · En��Ù��t��r,��,
TJ�t� =1
sin2 ��i�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�� ,
Hn�Ù��t��r,�� = ����−1f�t� � En
���t��r,��,
٠� ��, � �, n � �1,2�. �36�
n Eqs. (35) and (36), En���� ��r ,�� and En,z
��� ��r ,�� are theransverse and z components of En
�Ù�� ��r ,�� � =r , t�, re-pectively, rd= �x ,y ,z−d�T, f�r�=�−1�cos �i� , cos ��i� ,cos ��i��T is the spatial frequency of the reflected wave,
�t�=�II−1�cos �t� , cos ��t� , cos ��t��T is the spatial frequency of
he transmission wave, �II=� /nII is the wavelength in me-ium II, �cos �t� , cos ��t��=nII
−1�cos �i� , cos ��i��,��t� is the re-ractive angle in medium II determined by cos2 �t�
cos2 ��t�+cos2��t�=1, and TJ�r� and TJ�t� are determined byhe well-known Fresnel formula (Chap. 1.5 of Ref. 1).
Corresponding to E1�����i���1 ,� ;f1 ,f2�, E1�
����i���2 ,� ;f2�,
2�����i���1 ,� ;f1 ,f2�, and E2�
����i���2 ,� ;f2� in Eqs. (32), thelectric and magnetic vectors of the incident � = i�, re-ected � =r�, and transmission � = t� fields determinedy Eqs. (34)–(36) are denoted by �E1
���� ��r1 ,� ;f1 ,f2�,
1���� ��r1 ,� ;f1 ,f2��, �E1
���� ��r2 ,� ;f2�, H1���� ��r2 ,� ;f2��,
E2���� ��r1 ,� ;f1 ,f2�, H2
���� ��r1 ,� ;f1 ,f2��, andE2
���� ��r2 ,� ;f2�, H2���� ��r2 ,� ;f2��; respectively. By Eqs.
17), (19), and (21), the electric, magnetic, and mixedross-spectral density tensors of the incident � = i�, re-ected � =r�, and transmission � = t� waves are
WJ�e�� ��r1,r2,�� = E1���� ��r1,�;f1,f2�„E1
���� �
��r2,�;f2�…+d2f1d2f2 + E2���� �
��r1,�;f1,f2�„E2���� ��r2,�;f2�…+d2f1d2f2,
WJ�h�� ��r1,r2,�� = H1���� ��r1,�;f1,f2�„H1
���� �
��r2,�;f2�…+d2f1d2f2 + H2���� �
��r1,�;f1,f2�„H2���� ��r2,�;f2�…+d2f1d2f2,
WJ�m�� ��r,r2,�� = E1���� ��r1,�;f1,f2�„H1
���� �
��r2,�;f2�…+d2f1d2f2 + E2���� �
��r1,�;f1,f2�„H2���� ��r2,�;f2�…+d2f1d2f2.
�37�
. Comparisons and Remarkshe problem in Subsection 3.A can also be solved with theropagation theories based on SCMD, referring to therocedures in Subsection 3.B. The SCMD of the
��e��i���1 ,�2 ,�� in Subsection 3.A is
WJ��e��i���1,�2,�� = ��Axx
Ayx�exp�i2�f�i�T�1��
���1
0�exp�i2�f�i�T�2��+
���Axy
Ayy�exp�i2�f�i�T�1��
���0
1�exp�i2�f�i�T�2��+
, �38�
here �Axx Axy
Ayx Ayy �=AJ. SCMD (38) can be written out in a
traightforward way without solving the eigenvalue equa-ions, at the cost of more modes than CMD (25). The re-ectance R and the transmittance T of the numerical ex-mple in Subsection 3.A are calculated with theropagation theories based on SCMD, which are identicalith the results in Subsection 3.A, as shown by the “+”
urve in Fig. 2.Although the two examples in Section 3 have simple so-
utions with explicit expressions, the proposed propaga-ion theories are not limited to such simple cases. As men-ioned following Eqs. (4), the abstract operators PJ�e�
�r ,� ,�� and PJ�h��r ,� ,�� can represent any linear propa-ation physics governed by the linear Maxwell equations3). So the propagation theories in this paper are appli-able to any propagation problem of partially coherentlectromagnetic fields governed by the linear Maxwellquations (3), including those complex problems withoutolutions with explicit expressions, such as the rigorouslectromagnetic diffraction problem of subwavelengthratings4 illuminated by partially coherent electromag-etic fields.
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2216 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
. CONCLUSIONSropagation theories of partially coherent electromag-etic fields based on coherent mode decomposition (CMD)r separated-coordinate mode decomposition (SCMD) areroposed in this paper. The SCMD is a generalization ofhe CMD and can be obtained in a straightforward wayithout the need to solve the eigenvalue integral equa-
ions needed by the CMD, at the cost of more modes thanhe CMD. The proposed theories are applicable to anyropagation problem of partially coherent electromag-etic fields governed by the linear Maxwell equations.ased on the proposed theories, various powerful theorieseveloped for the propagation of fully coherent electro-agnetic fields are applicable to the propagation of par-
ially coherent electromagnetic fields. Two examples arerovided to illustrate the validity of the proposed theo-ies.
PPENDIX A: PROOF THAT PJ„e…„r,�,�) AND
„h…„r,�,�) ARE LINEAR TENSOR
PERATORSased on the linear property of the Maxwell equations (3)nd the equivalence between Eqs. (3) and (4), it can beasily proved that PJ�e��r ,� ,�� and PJ�h��r ,� ,�� are two lin-ar operators, namely,
PJ�e��r,�,���c1E�1�i� ��,�� + c2E�2
�i� ��,���
= c1PJ�e��r,�,��E�1�i� ��,�� + c2PJ�e��r,�,��E�2
�i� ��,��,
�A1a�
PJ�h��r,�,���c1E�1�i� ��,�� + c2E�2
�i� ��,���
= c1PJ�h��r,�,��E�1�i� ��,�� + c2PJ�h��r,�,��E�2
�i� ��,��,
�A1b�
here c1 and c2 are two arbitrary complex numbers and
�1�i� �� ,�� and E�2
�i� �� ,�� are two arbitrary E��i��� ,��. Let �
�x ,y ,z� denote the unit vector along the three coordi-ate axes; then Eq. (4a) can be rewritten as
E �r,�� = � · �PJ�e��r,�,��E��i���,���
= � · �PJ�e��r,�,���xEx�i���,�� + yEy
�i���,����
= � · �PJ�e��r,�,���xEx�i���,����
+ � · �PJ�e��r,�,���yEy�i���,����
= P x�e��r,�,��Ex
�i���,�� + P y�e��r,�,��Ey
�i���,��,
�A2a�
here the operator P ��e��r ,� ,�� is defined by
P ��e��r,�,��E�
�i���,�� = � · �PJ�e��r,�,����E��i���,����,
� � �x,y�. �A2b�
n Eq. (A2a), the centered dot means dot product, and theinear property (A1a) is applied to the third equality. The
inear property (A1a) shows that P ��e��r ,� ,��, defined by
q. (A2b), satisfies
��e��r,�,���c1E�,1
�i� ��,�� + c2E�,2�i� ��,���
= c1P ��e��r,�,��E�,1
�i� ��,�� + c2P ��e��r,�,��E�,2
�i� ��,��, �A3�
here c1 and c2 are two arbitrary complex numbers and
�,1�i� �� ,�� and E�,2
�i� �� ,�� are two arbitrary E��i��� ,��. Equa-
ion (A2a) shows that PJ�e��r ,� ,�� is a 3�2 tensor opera-or, and Eq. (A3) indicates that the six elements of PJ�e�
�r ,� ,�� are all linear operators. Based on the linearroperty (A1b), it can be similarly proved that PJ�h�
�r ,� ,�� is also a 3�2 tensor operator, with all of its sixlements being linear operators.
PPENDIX B: PROOF OF EQUALITY “5” INQUATIONS (11)et ��n��� �n=1,2, . . . � denote any complete functionroup defined on plane z=0, which can be of real value.he random electric component Ek
�i��� ,�� �k� �x ,y�� in Eq.11a) can be expressed as
Ek�i���,�� = �
n=1
�
ckn����n���, �B1�
here ckn��� is a complex random variable dependent on �n general. Then equality 5 in Eq. (11a) can be derived as
��k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*Ek�i���1,��„El
�i���2,��…*�=1��
k�
lPpk
�e��r1,�1,��„Pql�e��r2,�2,��…*�
m=1
�
ckm����m��1����
n=1
�
cln����n��2��*�=2��
k�
l�m=1
�
�n=1
�
ckm���cln* ���Ppk
�e��r1,�1,��
�„Pql�e��r2,�2,��…*�m��1��n��2��
=3
�k
�l
�m=1
�
�n=1
�
�ckm���cln* ����Ppk
�e��r1,�1,��
�„Pql�e��r2,�2,��…*�m��1��n��2�
=4
�k
�l
Ppk�e��r1,�1,��
�„Pql�e��r2,�2,��…* �
m=1
�
�n=1
�
�ckm���cln* �����m��1��n��2�
![Page 10: Propagation theories of partially coherent electromagnetic fields based on coherent or separated-coordinate mode decomposition](https://reader031.vdocuments.mx/reader031/viewer/2022020407/5750821f1a28abf34f96c26c/html5/thumbnails/10.jpg)
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=5
�k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*��m=1
�
ckm����m��1����
n=1
�
cln����n��2��*�=6
�k
�l
Ppk�e��r1,�1,��„Pql
�e��r2,�2,��…*�Ek�i���1,��„El
�i���2,��…*�,
�B2�
here Eq. (B1) is used for equalities 1 and 6, the linear�e�
roperty of operator Ppk�r1 ,�1 ,�� is applied to equalities 2 TwtAJ
W
E
nd 4, and the linear property of the ensemble averageperator � � is used for equalities 3 and 5.
PPENDIX C: SOLUTION OF SEPARATED-OORDINATE MODE DECOMPOSITON. In the Form of a Serieset ��n��� �n=1,2, . . . � denote any complete functionroup defined on plane z=0, which can be of real value.
J�e��i�
hen W� ��1 ,�2 ,�� can be expressed asWJ��e��i���1,�2,�� = Wxx
�e��i���1,�2,�� Wxy�e��i���1,�2,��
„Wxy�e��i���2,�1,��…* Wyy
�e��i���1,�2,��� = �m=1
�
�n=1
�
cmn�xx�����m��1��n��2� �
m=1
�
�n=1
�
cmn�xy�����m��1��n��2�
�m=1
�
�n=1
�
�cmn�xy�����*�m��2��n��1� �
m=1
�
�n=1
�
cmn�yy�����m��1��n��2��
= �m=1
�
�n=1
�
cmn�xx������m��1�
0 ���n��2� 0� + �m=1
�
�n=1
�
cmn�xy������m��1�
0 ��0 �n��2�� + �m=1
�
�n=1
�
cmn�xy����*� 0
�n��1�����m��2� 0� + �
m=1
�
�n=1
�
cmn�yy����� 0
�m��1���0 �n��2��, �C1�
here cmn�xx����, cmn
�xy����, and cmn�yy���� are complex coefficients.
quation (C1) is just a SCMD in the form of a series.
. In the Form of an Integralhe transverse component E�
�i��� ,�� of the incident ran-om electric field on plane z=0 can be expressed as a su-erposition of planar waves of (Chap. 3.2 of Ref. 2)
E��i���,�� = A�
�e��f,��exp�i2�fT��d2f, �C2�
here f= �fx , fy ,0�T is the transverse spatial frequency ofach planar wave component and
A��e��f,�� = E�
�i���,��exp�− i2�fT��d2� �C3�
s called the angular spectrum of E��i��� ,��. The insertion
f Eq. (C2) into definition (2) yields
WJ��e��i���1,�2,�� =� A�
�e��f1,��exp�i2�f1T�1�d2f1�
� A��e��f2,��exp�i2�f2
T�2�d2f2�+�= AJ�
�e��f1,f2,��exp�i2��f1T�1
− f2T�2��d2f1d2f2, �C4�
here AJ��e��f1 ,f2 ,��= �A�
�e��f1 ,��„A��e��f2 ,�…�+� is called the
ransverse electric angular spectrum correlation tensor.11
��e��f1 ,f2 ,�� can be written as
AJ��e��f1,f2,�� = Axx
�e��f1,f2,�� Axy�e��f1,f2,��
Ayx�e��f1,f2,�� Ayy
�e��f1,f2,���= �Axx
�e��f1,f2,��
Ayx�e��f1,f2,����1 0� + �Axy
�e��f1,f2,��
Ayy�e��f1,f2,����0 1�.
�C5�
ith Eq. (C5) inserted into it, Eq. (C4) becomes
WJ��e��i���1,�2,�� = ��Axx
�e��f1,f2,��
Ayx�e��f1,f2,���exp�i2�f1
T�1����1
0�exp�i2�f2T�2��+
d2f1d2f2
+ ��Axy�e��f1,f2,��
Ayy�e��f1,f2,���
�exp�i2�f1T�1��
���0
1�exp�i2�f2T�2��+
d2f1d2f2. �C6�
quation (C6) is just a SCMD in the form of an integral.
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2218 J. Opt. Soc. Am. A/Vol. 23, No. 9 /September 2006 Liu et al.
CKNOWLEDGMENTShis research was supported by the Natural Scienceoundation of Tianjin under grant 06YFJMJC01500, byhe Open Research Fund of the Key Laboratory of Opto-lectronic Information Science and Technology of the Edu-ation Ministry of China under grant 2005-04, and by theund for the Development Project of Science and Technol-gy of Tianjin under grant 043103011.
Corresponding author Haitao Liu can be reached by-mail, [email protected]; phone; 8622 23506422; orax; 8622 23502275.
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