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The Mueller matrix formalism in diffraction phenomena
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
THE MUELLER MATRIX FORMALISM IN DIFFRACTION PHENOMENA A. DOMAIQSKI Institute o f Physics, Technical University, Warsaw, Poland *
Received 3 April 1975
Electromagnetic diffraction at the edge of a plane screen is described by the use of the Mueller matrix. The transformation law is found for the Stokes parameters of the incident and diffracted wave by applying the MueUer method to several cases of diffraction.
1. Introduction The aim of this paper is to apply the Mueller matrix in the description of electromagnetic diffraction. The Mueller matrix may be found from the relation: [S"] = [M] [S'] ,
(1)
where IS'] are the Stokes parameters of the incident wave and [S"] are the Stokes parameters of the diffracted wave We find [S'] and [S"] from Wolf's coherence matrix of the incident wave [ J ' ] and o f the diffracted wave [ J " ] , respectively, from the relation:
S1 = S2 S3
I:°° il[xl 0 0
1 1
-i
i
-
Jxy
.
(2)
Jyx Jyy
The coherence matrix of the arbitrarily polarized incident wave [ J ' ] is very simple, but the coherence matrix of the diffracted wave [J"] can only be found once the solution of the diffraction problem is known. Karczewski and Wolf [1 ] have found the coherence matriees o f the diffracted wave at an aperture in a plane screen (for "magnetic", "electric" and "electromagnetic" screens). Moreover we know the coherence matrices for a perfectly conducting half-plane [2] and for a half-plane of finite conductance [3]. The coherence matrices for the above cases were found for a monochromatic incident wave (and also for a quasimonochromatic incident wave in the case of a perfectly conducting half-plane). The diffracted wave was analyzed in the zone of shadow at large distances from the diffracting edge.
2. The Mueller matrices for the "magnetic", "electric" and "electromagnetic" screens The Stokes parameters of the incident wave for diffraction at the "magnetic", "electric" and "electromagnetic" * Mailing address: Instytut Fizyki, Politecbnika Warszawska, 00-662 Warszawa, Koszykowa 75, Poland. 281
Volume 14, number 3
OPTICS COMMUNICATIONS
>
July 1975
~'/%Z' 2q
B" i!ili¸¸i Fig. 1. Orientation of the vectors K, L, M, k, n, R, r0, r. screens take the form: S(
-IAL[2+ JAM j2
S
lAg 12- IAM 12 =
(3)
,
S:
I| 2 JALI [AM[ cos~
S
[2 IAL[ IAMI sin8
where A L and A M are the component of the electric vector in the direction L and M, respectively (fig. 1) and 8 is the phase difference between A L and A M . The Stokes parameters of the diffracted wave in each of the approximate cases may be found from the corresponding coherence matrix [1]. The Stokes parameters of the diffracted wave at the "magnetic" screen (n X E = n X E i at the aperture A and n X E = 0 at the unlighted side B- of the screen) are of the form: S0m
a2m]AL [2 + (bl2m + b2m)IAM j2 + 2almblm[AL[ [A M ] cos8
S1"m a2m[AL[2+(b2m_. b2m 2 ) JAM [2 + 2almblmlALI IAMI cos 8 ,, = 4IF[ 2 S2m , 2blm b2m JAM i2 + 2aim b2m IALI IA M [ cos 8 [_S~m
'
(4)
2aim b2m IALI JAM[ sin 8
where
(s)
F = (iko/4rr) [exp (ikoro)/ro] "n f / e x p [ik0(K - k ) " R ] ds , A with the vectors n, k, R, r 0, r shown in fig. 1, k 0 = oo/c, and alm=-Kz[(1
k2)/(1--K.~)l 1/2,
blm = - kxky [(1 - Ky2)/(1- ky2)] I/2+ KxKy [ ( 1 - k2)/(1 - K 2 ) ] 1/2,
(6) b2m = - k z [(1 - K2)/(1 - ky2)]1/2.
The Mueller matrix for the "magnetic" screen takes the form: ~a2m + bl2m + b2m
al2m - b2m
~a2m+b2m
a2m
['Mm]=21FI212blmb2m 282
b2m
b2m
2almblm
0
b2m +b2m
2almblm
0
-2blmb2m
2almb2m
0
0
0
2alto b2m
(7)
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
The Stokes parameters of the diffracted wave at the "electric" screen (n × H = n X H i at the aperture A and n X H = 0 at the unlighted side B - of the screen) are of the form:
iS"/le
(ale2 _ a2e) IAL 12 - b2elAM 12 -
= 4 IFI 2
2a2eb2elALI IAMI cos6
(8)
2alea2elAL 12+2aleb2elAL I [AM I cos5
L;i:t
where
2a2eb2elAL ] JAM ] cos8
(a2e + a22e)IAL 12+ b2eLAM 12 +
0e
2aleb2elALI IAM[ sin5
ale = - k z [(1 - Ky2)/(1 - ky2)] 1/2, (9)
a2e=kxky [(1- K2)/(1-k2)] 1/2- KxKy[(1-g)/(1-Ky)]2
1/2,
b2e =
- K z [(1 -- k2)/(1
- K 2 ) I 1/2
The Mueller matrix for the "electric" screen takes the form: -a21e+ U2e-2+ b2e
a2e + a2e - b2e
2a2eb2e
0
a 2 e - a 2 e - b2e
a2e - ,2 e+ b2e
-2a2eb2e
0
2alea2e
2alea2e
2aleb2e
0
0
0
0
2aleb2,
(10)
[Me] = 2 IFI 2
The Stokes parameters of the diffracted wave at the "electromagnetic" screen (n × E = n X E i and n X H = n × H i at the aperture A as well as n X E = 0 and n X H = 0 at the unlighted side B - of the screen) are of the form:
-,, S0em ,, Slem
-
2
(alem
+
2
a2em)lALI
2+
2
(blem
+
2
b2em) lAMI
2
+2(alemblem+a2emblem)]Ak]lAMIcos8-] |
S~em
2 2 2+ (blem-b2em)lAMI 2 2 2+ 2(alemblem _ a2emb2em)lALI]AMI cosS[ (alem-a2em)lALI 2alemb2em[ALi2+ 2blemb2emlAMi2+ 2(alemb2em+a2emblem)lALiiAMiCOS8 I '
S~em_
2 (ale m b2e m - a2e m b2e m ) [Z L I IA M I sin 8
=4tFI2
]
(11) where ale m = (alm + ale)/2,
a2e m = a2e/2,
ble m = b lm/2,
b2e m = (b2m + b2e)/2.
(12)
The Mueller matrix for the "electromagnetic" screen can be written in the form: [Mem I = 2 IFI 2 2 2 2 + 2 -2 -2 a2em + a2em + ble m + b2e m - ale m a2em-Olem-O2em X
a l2e m - a 2 2e m +b2em - b2em
2 2 -2 + -°2era 2 alem-a2em-°lem
2(alemblem+a2emb2em) 0 2(alemblem-a2emb2em) 0
2(alema2e m + blemb2e m )
2 ( a l e m a 2 e m - b lemb2em)
2(alemb2e m + a2emblem)
0
0
0
0
2 (a lemb2em - a2em b le'm (13)
It is evident that the matrix elements depend on the directions of the incident and diffracted waves, on the distance from the screen and on the kind of the diffracting screen (the parameters a and b depend on the boundary conditions on the screen). 283
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
3. The Muellermatrices for diffraction at conducting half-planes The coherence matrix of the a wave diffracted on an ideally conducting half-plane was found by Jannson [2]. He started from the Sommerfeld solution of the diffraction problem. The modified Sommerfeld solution due to Raman and Krishnan [4] was used to find the coherence matrix of the diffracted wave at a half-plane of finite conductivity [3]. The Stokes ~arameters of the incident wave are of the same form in the two cases:
s; Si'=" t
$2
i
_$3j
a2L a 2
'
(14)
2aLa M c o s A ~ _2aLaN sin Aft'
where a L and a N are the component of the electric vector amplitude in the direction L and M, respectively (fig. 2), and A ~ is the phase difference between a L and a M . The Stokes parameters of the diffracted wave at a half-plane of finite conductivity (in the more general case) take the form:
~
'St
/
t-
r2aM 22
r¢
= rl a L
Slr ft
22
r2aM
2aLaMrlr 2 cos (AqJ + A¢I
S2r
LS;r
2aLa M r 1 r 2 sin ( A ~ + A¢) L_
where rl
COS½(qb--'I'0) + COS{ (q~+¢0)
2(27rk,o)1/2 r2 -
1
{i
1
2(2rrkp)l/2 , cos½(qb--qb0)
Dp
/
+ [COS½('I'+¢0)]2 j
Cs 12 D2 }t/2 COS{(dp+qb0) + [cos{(qb+~b0)] 2 '
(16)
Dp cos { (qb -- qb0) D s cos { (qb -- qb0) A¢ = arctg Cp cos 1 (qb - - * 0 ) + cos 1 (d# + qb0) -- arc tg Cs cos ½(q~ -qb0) - cos ½ (qb + qb0) ' with p the distance from the diffracting edge, and q50 and qb the angle of incidence and diffraction, respectively (fig. 2). R s = Cs+ iD s and Rp = Cp + iDp are the reflection coefficients for parallel and perpendicular polarization, respectively.
Fig. 2. Definition of: angle of incidence ~o, angle of diffraction • , and the vectors p, K, L, M. 284
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
The Mueller matrix for the above case can be written in the form:
-(r2+r2)/2
(r2-r2)/2
0
0
(r 2-r2)/2
(r 2+ r2)/2
0
0
0
0
r 1 r 2 cos A~o
-rlr 2 sinA~
0
0
r 1 r 2 sin A~o
rlr 2 cos A¢
[Mr] =
(17)
The Mueller matrix for a perfectly conducting half-plane is simpler because Rs= Rp = 1 and thus:
[Mi] =
(r2i+r22i)/2
(r21i-r2i)/2
0
0
(r21i-r2i)/2
(r2i+r22i)/2
0
0
0
0
rlir2i
0
0
0
0
rlir2i
(18)
where
'
1 .... I 1 Jr rli=2(27rko)l/2 cosa(qb--qbO) cos ½(~+q50
'
r2i-
2(27rkp)l/2
Ecos ½(qb-1
1
qb0)
q
cos ½ (,I,+~0)J ' (19)
A~=O. The elements of the Mueller matrix for the conducting screens depend on the angles of incidence and diffraction, on the distance from the edge of the screen and on the material of the screen (reflection factors depend on the conductivity).
4. Conclusions The application of the Mueller matrix in the description of the electromagnetic diffraction shows some advantages: a) The whole problem of far-field diffraction may be described by one matrix relation: [S"] = [M] IS']. b) The above relation reveals that the properties of the diffracted wave depend on the properties of the incident wave (described by the Stokes parameters), on the geometry of diffraction, and on the screen material (described by the Mueller matrix). c) Suppose that the wave passes through optical devices for which we know the Mueller matrices [M 1 ], [M2] ..... [Mn], and that it is subsequently diffracted. In this case we can analyze the diffracted wave in the shadow zone by means of the relation: [S"] = [M]
[Mn] ... [M2]
[1141] [S'] ,
(20)
where [M] is the Mueller matrix for diffraction. d) We can analyze states of polarization of the diffracted wave by means of the relation [S"] = [M] [S'], because the polarization ratio, the ratio of the axes of the ellipse, and the inclination of the axes to the incident plane depend directly on the Stokes parameters [5]. e) We can analyze the properties of the incident wave from the relation: [S'] = [ M ] - I I S " ] ,
(21)
where [M] -1 is the inverse Mueller matrix, which may be found because in the above cases the Mueller matrices are nonsingular. 285
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
f) An analysis of the states of polarization of the diffracted wave can be made on the basis of the normalized Poincar6 sphere [2]. Every point of this sphere is defined by the coordinates S 1/S 0, $2/S0, $3/S 0 which can be found directly from the relation IS"] = [114] [S']. Finally we would like to remark that the elements of the Mueller matrix depend in a highly involved manner on the screen properties and on the geometry of diffraction, and attempts to separate the influence of the screen properties from the influence of the geometry have failed.
References [1] [2] [3] [4] [5]
286
B. Karczewski, E. Wolf, J. Opt. Soc. 56 (1966) 1207. T. Jannson, Acta Phys. Polon. 36 (1969) 803. A. Domafiski, Acta Phys. Polon. 40 (1971) 765. C.V. Raman, K.S. Krishnan, Proc. Roy. Soc. (A) (London) 116 (1927). M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970).
OPTICS COMMUNICATIONS
July 1975
THE MUELLER MATRIX FORMALISM IN DIFFRACTION PHENOMENA A. DOMAIQSKI Institute o f Physics, Technical University, Warsaw, Poland *
Received 3 April 1975
Electromagnetic diffraction at the edge of a plane screen is described by the use of the Mueller matrix. The transformation law is found for the Stokes parameters of the incident and diffracted wave by applying the MueUer method to several cases of diffraction.
1. Introduction The aim of this paper is to apply the Mueller matrix in the description of electromagnetic diffraction. The Mueller matrix may be found from the relation: [S"] = [M] [S'] ,
(1)
where IS'] are the Stokes parameters of the incident wave and [S"] are the Stokes parameters of the diffracted wave We find [S'] and [S"] from Wolf's coherence matrix of the incident wave [ J ' ] and o f the diffracted wave [ J " ] , respectively, from the relation:
S1 = S2 S3
I:°° il[xl 0 0
1 1
-i
i
-
Jxy
.
(2)
Jyx Jyy
The coherence matrix of the arbitrarily polarized incident wave [ J ' ] is very simple, but the coherence matrix of the diffracted wave [J"] can only be found once the solution of the diffraction problem is known. Karczewski and Wolf [1 ] have found the coherence matriees o f the diffracted wave at an aperture in a plane screen (for "magnetic", "electric" and "electromagnetic" screens). Moreover we know the coherence matrices for a perfectly conducting half-plane [2] and for a half-plane of finite conductance [3]. The coherence matrices for the above cases were found for a monochromatic incident wave (and also for a quasimonochromatic incident wave in the case of a perfectly conducting half-plane). The diffracted wave was analyzed in the zone of shadow at large distances from the diffracting edge.
2. The Mueller matrices for the "magnetic", "electric" and "electromagnetic" screens The Stokes parameters of the incident wave for diffraction at the "magnetic", "electric" and "electromagnetic" * Mailing address: Instytut Fizyki, Politecbnika Warszawska, 00-662 Warszawa, Koszykowa 75, Poland. 281
Volume 14, number 3
OPTICS COMMUNICATIONS
>
July 1975
~'/%Z' 2q
B" i!ili¸¸i Fig. 1. Orientation of the vectors K, L, M, k, n, R, r0, r. screens take the form: S(
-IAL[2+ JAM j2
S
lAg 12- IAM 12 =
(3)
,
S:
I| 2 JALI [AM[ cos~
S
[2 IAL[ IAMI sin8
where A L and A M are the component of the electric vector in the direction L and M, respectively (fig. 1) and 8 is the phase difference between A L and A M . The Stokes parameters of the diffracted wave in each of the approximate cases may be found from the corresponding coherence matrix [1]. The Stokes parameters of the diffracted wave at the "magnetic" screen (n X E = n X E i at the aperture A and n X E = 0 at the unlighted side B- of the screen) are of the form: S0m
a2m]AL [2 + (bl2m + b2m)IAM j2 + 2almblm[AL[ [A M ] cos8
S1"m a2m[AL[2+(b2m_. b2m 2 ) JAM [2 + 2almblmlALI IAMI cos 8 ,, = 4IF[ 2 S2m , 2blm b2m JAM i2 + 2aim b2m IALI IA M [ cos 8 [_S~m
'
(4)
2aim b2m IALI JAM[ sin 8
where
(s)
F = (iko/4rr) [exp (ikoro)/ro] "n f / e x p [ik0(K - k ) " R ] ds , A with the vectors n, k, R, r 0, r shown in fig. 1, k 0 = oo/c, and alm=-Kz[(1
k2)/(1--K.~)l 1/2,
blm = - kxky [(1 - Ky2)/(1- ky2)] I/2+ KxKy [ ( 1 - k2)/(1 - K 2 ) ] 1/2,
(6) b2m = - k z [(1 - K2)/(1 - ky2)]1/2.
The Mueller matrix for the "magnetic" screen takes the form: ~a2m + bl2m + b2m
al2m - b2m
~a2m+b2m
a2m
['Mm]=21FI212blmb2m 282
b2m
b2m
2almblm
0
b2m +b2m
2almblm
0
-2blmb2m
2almb2m
0
0
0
2alto b2m
(7)
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
The Stokes parameters of the diffracted wave at the "electric" screen (n × H = n X H i at the aperture A and n X H = 0 at the unlighted side B - of the screen) are of the form:
iS"/le
(ale2 _ a2e) IAL 12 - b2elAM 12 -
= 4 IFI 2
2a2eb2elALI IAMI cos6
(8)
2alea2elAL 12+2aleb2elAL I [AM I cos5
L;i:t
where
2a2eb2elAL ] JAM ] cos8
(a2e + a22e)IAL 12+ b2eLAM 12 +
0e
2aleb2elALI IAM[ sin5
ale = - k z [(1 - Ky2)/(1 - ky2)] 1/2, (9)
a2e=kxky [(1- K2)/(1-k2)] 1/2- KxKy[(1-g)/(1-Ky)]2
1/2,
b2e =
- K z [(1 -- k2)/(1
- K 2 ) I 1/2
The Mueller matrix for the "electric" screen takes the form: -a21e+ U2e-2+ b2e
a2e + a2e - b2e
2a2eb2e
0
a 2 e - a 2 e - b2e
a2e - ,2 e+ b2e
-2a2eb2e
0
2alea2e
2alea2e
2aleb2e
0
0
0
0
2aleb2,
(10)
[Me] = 2 IFI 2
The Stokes parameters of the diffracted wave at the "electromagnetic" screen (n × E = n X E i and n X H = n × H i at the aperture A as well as n X E = 0 and n X H = 0 at the unlighted side B - of the screen) are of the form:
-,, S0em ,, Slem
-
2
(alem
+
2
a2em)lALI
2+
2
(blem
+
2
b2em) lAMI
2
+2(alemblem+a2emblem)]Ak]lAMIcos8-] |
S~em
2 2 2+ (blem-b2em)lAMI 2 2 2+ 2(alemblem _ a2emb2em)lALI]AMI cosS[ (alem-a2em)lALI 2alemb2em[ALi2+ 2blemb2emlAMi2+ 2(alemb2em+a2emblem)lALiiAMiCOS8 I '
S~em_
2 (ale m b2e m - a2e m b2e m ) [Z L I IA M I sin 8
=4tFI2
]
(11) where ale m = (alm + ale)/2,
a2e m = a2e/2,
ble m = b lm/2,
b2e m = (b2m + b2e)/2.
(12)
The Mueller matrix for the "electromagnetic" screen can be written in the form: [Mem I = 2 IFI 2 2 2 2 + 2 -2 -2 a2em + a2em + ble m + b2e m - ale m a2em-Olem-O2em X
a l2e m - a 2 2e m +b2em - b2em
2 2 -2 + -°2era 2 alem-a2em-°lem
2(alemblem+a2emb2em) 0 2(alemblem-a2emb2em) 0
2(alema2e m + blemb2e m )
2 ( a l e m a 2 e m - b lemb2em)
2(alemb2e m + a2emblem)
0
0
0
0
2 (a lemb2em - a2em b le'm (13)
It is evident that the matrix elements depend on the directions of the incident and diffracted waves, on the distance from the screen and on the kind of the diffracting screen (the parameters a and b depend on the boundary conditions on the screen). 283
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
3. The Muellermatrices for diffraction at conducting half-planes The coherence matrix of the a wave diffracted on an ideally conducting half-plane was found by Jannson [2]. He started from the Sommerfeld solution of the diffraction problem. The modified Sommerfeld solution due to Raman and Krishnan [4] was used to find the coherence matrix of the diffracted wave at a half-plane of finite conductivity [3]. The Stokes ~arameters of the incident wave are of the same form in the two cases:
s; Si'=" t
$2
i
_$3j
a2L a 2
'
(14)
2aLa M c o s A ~ _2aLaN sin Aft'
where a L and a N are the component of the electric vector amplitude in the direction L and M, respectively (fig. 2), and A ~ is the phase difference between a L and a M . The Stokes parameters of the diffracted wave at a half-plane of finite conductivity (in the more general case) take the form:
~
'St
/
t-
r2aM 22
r¢
= rl a L
Slr ft
22
r2aM
2aLaMrlr 2 cos (AqJ + A¢I
S2r
LS;r
2aLa M r 1 r 2 sin ( A ~ + A¢) L_
where rl
COS½(qb--'I'0) + COS{ (q~+¢0)
2(27rk,o)1/2 r2 -
1
{i
1
2(2rrkp)l/2 , cos½(qb--qb0)
Dp
/
+ [COS½('I'+¢0)]2 j
Cs 12 D2 }t/2 COS{(dp+qb0) + [cos{(qb+~b0)] 2 '
(16)
Dp cos { (qb -- qb0) D s cos { (qb -- qb0) A¢ = arctg Cp cos 1 (qb - - * 0 ) + cos 1 (d# + qb0) -- arc tg Cs cos ½(q~ -qb0) - cos ½ (qb + qb0) ' with p the distance from the diffracting edge, and q50 and qb the angle of incidence and diffraction, respectively (fig. 2). R s = Cs+ iD s and Rp = Cp + iDp are the reflection coefficients for parallel and perpendicular polarization, respectively.
Fig. 2. Definition of: angle of incidence ~o, angle of diffraction • , and the vectors p, K, L, M. 284
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
The Mueller matrix for the above case can be written in the form:
-(r2+r2)/2
(r2-r2)/2
0
0
(r 2-r2)/2
(r 2+ r2)/2
0
0
0
0
r 1 r 2 cos A~o
-rlr 2 sinA~
0
0
r 1 r 2 sin A~o
rlr 2 cos A¢
[Mr] =
(17)
The Mueller matrix for a perfectly conducting half-plane is simpler because Rs= Rp = 1 and thus:
[Mi] =
(r2i+r22i)/2
(r21i-r2i)/2
0
0
(r21i-r2i)/2
(r2i+r22i)/2
0
0
0
0
rlir2i
0
0
0
0
rlir2i
(18)
where
'
1 .... I 1 Jr rli=2(27rko)l/2 cosa(qb--qbO) cos ½(~+q50
'
r2i-
2(27rkp)l/2
Ecos ½(qb-1
1
qb0)
q
cos ½ (,I,+~0)J ' (19)
A~=O. The elements of the Mueller matrix for the conducting screens depend on the angles of incidence and diffraction, on the distance from the edge of the screen and on the material of the screen (reflection factors depend on the conductivity).
4. Conclusions The application of the Mueller matrix in the description of the electromagnetic diffraction shows some advantages: a) The whole problem of far-field diffraction may be described by one matrix relation: [S"] = [M] IS']. b) The above relation reveals that the properties of the diffracted wave depend on the properties of the incident wave (described by the Stokes parameters), on the geometry of diffraction, and on the screen material (described by the Mueller matrix). c) Suppose that the wave passes through optical devices for which we know the Mueller matrices [M 1 ], [M2] ..... [Mn], and that it is subsequently diffracted. In this case we can analyze the diffracted wave in the shadow zone by means of the relation: [S"] = [M]
[Mn] ... [M2]
[1141] [S'] ,
(20)
where [M] is the Mueller matrix for diffraction. d) We can analyze states of polarization of the diffracted wave by means of the relation [S"] = [M] [S'], because the polarization ratio, the ratio of the axes of the ellipse, and the inclination of the axes to the incident plane depend directly on the Stokes parameters [5]. e) We can analyze the properties of the incident wave from the relation: [S'] = [ M ] - I I S " ] ,
(21)
where [M] -1 is the inverse Mueller matrix, which may be found because in the above cases the Mueller matrices are nonsingular. 285
Volume 14, number 3
OPTICS COMMUNICATIONS
July 1975
f) An analysis of the states of polarization of the diffracted wave can be made on the basis of the normalized Poincar6 sphere [2]. Every point of this sphere is defined by the coordinates S 1/S 0, $2/S0, $3/S 0 which can be found directly from the relation IS"] = [114] [S']. Finally we would like to remark that the elements of the Mueller matrix depend in a highly involved manner on the screen properties and on the geometry of diffraction, and attempts to separate the influence of the screen properties from the influence of the geometry have failed.
References [1] [2] [3] [4] [5]
286
B. Karczewski, E. Wolf, J. Opt. Soc. 56 (1966) 1207. T. Jannson, Acta Phys. Polon. 36 (1969) 803. A. Domafiski, Acta Phys. Polon. 40 (1971) 765. C.V. Raman, K.S. Krishnan, Proc. Roy. Soc. (A) (London) 116 (1927). M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1970).