Abstract

Various aspects of the physics of partially polarized waves are discussed with applications to optical (lidar) reception problems. We focus on the issue of the optimal intensity reception of partially polarized waves scattered off a fluctuating object (ensemble of scatterers) of known polarization properties (measured Mueller matrix). Expressions for total available intensity and adjustable (polarization-dependent) intensity are derived in a clear and novel manner by using the coherency matrix approach. A general numerical technique is developed and illustrated for the optimization of adjustable intensity as a function of transmitted polarization. Closed-form expressions are derived for two important subcases, and numerical illustrations for the general case are discussed in detail, including the use of relevant experimental data.

© 1988 Optical Society of America

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  1. A nonmonochromatic (time-dependent) wave can be completely polarized only if the two components fluctuate in phase, i.e., if the polarization ellipse fluctuates in magnitude without changing shape.
  2. M. Born, E. Wolf, Principles of Optics, 2nd revised ed. (Pergamon, New York, 1964).
  3. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Pergamon, New York, 1960).
  4. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  6. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  7. W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
    [CrossRef]
  8. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  9. R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
    [CrossRef]
  10. R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1983).
    [CrossRef]
  11. H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–663 (1948).
  12. Introduction of [M] as a linear operator implies additivity of the Stokes vectors, which, in turn, implies incoherent addition of waves (see, for example, Ref. 4, p. 149).
  13. A. B. Kostinski, W.-M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas Propag. AP-341395–1404 (1986);H. Mieras, “Comment,” IEEE Trans. Antennas Propag. AP-341470–1471 (1986);A. B. Kostinski, W.-M. Boerner, “Reply,” IEEE Trans. Antennas Propag. AP-341471–1473 (1986).
    [CrossRef]
  14. Note that the range of Θ in Eq. (8a) is (−π/2, +π/2) rather than (0, π), which is in agreement with Ref. 4 but not with Ref. 2. Whereas (0, π) seems to be a convenient choice for the Poincaré sphere representation,2the range of (−π/2, +π/2) is used by all computers for the calculation of the inverse tangent. It is particularly important for the correct inversion of Eqs. (8a) and (8b) because 2Θ rather than Θ enters the Stokes parameters resulting in a degeneracy of Θ ± π/2.
  15. H. C. Ko, “On the reception of quasi-monochromatic, partially polarized radio waves,” Proc. IRE 50, 1950–1957 (1962).
    [CrossRef]
  16. G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).
  17. G. E. Schilov, Linear Algebra (Dover, New York, 1977).
  18. R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. (to be published).
  19. E. S. Fry, G. W. Kattawar, “Relationships between the elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef] [PubMed]
  20. B. J. Howell, “Measurement of the polarization effects of an instrument using partially polarized light,” Appl. Opt. 18, 809–812 (1979).
    [CrossRef] [PubMed]
  21. During the preparation of our paper we came across a discussion of a similar problem in the paper by van Zyl et al. [J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987)], which is concerned with the optimization of total intensity rather than adjustable intensity. Although the basic equations and physical reasoning are not entirely clear to us, the resulting mathematical and numerical problems turn out to be quite similar (e.g., the sixth-order equation) and provide an interesting comparison of methods.

1987 (1)

1986 (1)

A. B. Kostinski, W.-M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas Propag. AP-341395–1404 (1986);H. Mieras, “Comment,” IEEE Trans. Antennas Propag. AP-341470–1471 (1986);A. B. Kostinski, W.-M. Boerner, “Reply,” IEEE Trans. Antennas Propag. AP-341471–1473 (1986).
[CrossRef]

1985 (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

1983 (1)

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1983).
[CrossRef]

1981 (2)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

E. S. Fry, G. W. Kattawar, “Relationships between the elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
[CrossRef] [PubMed]

1979 (1)

1962 (1)

H. C. Ko, “On the reception of quasi-monochromatic, partially polarized radio waves,” Proc. IRE 50, 1950–1957 (1962).
[CrossRef]

1948 (1)

H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–663 (1948).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bailey, W. M.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Barakat, R.

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. (to be published).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bickel, W. S.

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Boerner, W.-M.

A. B. Kostinski, W.-M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas Propag. AP-341395–1404 (1986);H. Mieras, “Comment,” IEEE Trans. Antennas Propag. AP-341470–1471 (1986);A. B. Kostinski, W.-M. Boerner, “Reply,” IEEE Trans. Antennas Propag. AP-341471–1473 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd revised ed. (Pergamon, New York, 1964).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Fry, E. S.

Howell, B. J.

Kattawar, G. W.

Kim, K.

Ko, H. C.

H. C. Ko, “On the reception of quasi-monochromatic, partially polarized radio waves,” Proc. IRE 50, 1950–1957 (1962).
[CrossRef]

Kostinski, A. B.

A. B. Kostinski, W.-M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas Propag. AP-341395–1404 (1986);H. Mieras, “Comment,” IEEE Trans. Antennas Propag. AP-341470–1471 (1986);A. B. Kostinski, W.-M. Boerner, “Reply,” IEEE Trans. Antennas Propag. AP-341471–1473 (1986).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Pergamon, New York, 1960).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Pergamon, New York, 1960).

Mandel, L.

Mueller, H.

H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–663 (1948).

Schilov, G. E.

G. E. Schilov, Linear Algebra (Dover, New York, 1977).

Simon, R.

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1983).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wolf, E.

Am. J. Phys. (1)

W. S. Bickel, W. M. Bailey, “Stokes vectors, Mueller matrices, and polarized scattered light,” Am. J. Phys. 53, 468–478 (1985).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

A. B. Kostinski, W.-M. Boerner, “On foundations of radar polarimetry,” IEEE Trans. Antennas Propag. AP-341395–1404 (1986);H. Mieras, “Comment,” IEEE Trans. Antennas Propag. AP-341470–1471 (1986);A. B. Kostinski, W.-M. Boerner, “Reply,” IEEE Trans. Antennas Propag. AP-341471–1473 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

H. Mueller, “The foundations of optics,” J. Opt. Soc. Am. 38, 661–663 (1948).

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

R. Barakat, “Bilinear constraints between elements of the 4 × 4 Mueller–Jones transfer matrix of polarization theory,” Opt. Commun. 38, 159–161 (1981).
[CrossRef]

R. Simon, “The connection between Mueller and Jones matrices of polarization optics,” Opt. Commun. 42, 293–297 (1983).
[CrossRef]

Proc. IRE (1)

H. C. Ko, “On the reception of quasi-monochromatic, partially polarized radio waves,” Proc. IRE 50, 1950–1957 (1962).
[CrossRef]

Other (12)

G. Strang, Linear Algebra and Its Applications (Academic, New York, 1976).

G. E. Schilov, Linear Algebra (Dover, New York, 1977).

R. Barakat, “Conditions for the physical realizability of polarization matrices characterizing passive systems,” J. Mod. Opt. (to be published).

Introduction of [M] as a linear operator implies additivity of the Stokes vectors, which, in turn, implies incoherent addition of waves (see, for example, Ref. 4, p. 149).

Note that the range of Θ in Eq. (8a) is (−π/2, +π/2) rather than (0, π), which is in agreement with Ref. 4 but not with Ref. 2. Whereas (0, π) seems to be a convenient choice for the Poincaré sphere representation,2the range of (−π/2, +π/2) is used by all computers for the calculation of the inverse tangent. It is particularly important for the correct inversion of Eqs. (8a) and (8b) because 2Θ rather than Θ enters the Stokes parameters resulting in a degeneracy of Θ ± π/2.

A nonmonochromatic (time-dependent) wave can be completely polarized only if the two components fluctuate in phase, i.e., if the polarization ellipse fluctuates in magnitude without changing shape.

M. Born, E. Wolf, Principles of Optics, 2nd revised ed. (Pergamon, New York, 1964).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Pergamon, New York, 1960).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

During the preparation of our paper we came across a discussion of a similar problem in the paper by van Zyl et al. [J. van Zyl, C. H. Papas, and C. Elachi, “On the optimum polarizations of incoherently reflected waves,” IEEE Trans. Antennas Propag. AP-35, 818–825 (1987)], which is concerned with the optimization of total intensity rather than adjustable intensity. Although the basic equations and physical reasoning are not entirely clear to us, the resulting mathematical and numerical problems turn out to be quite similar (e.g., the sixth-order equation) and provide an interesting comparison of methods.

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Figures (1)

Fig. 1
Fig. 1

Polarization dependence of the adjustable intensity. The general topology of the surface is determined by the six stationary points corresponding to the six roots of Eq. (20) with the global maximum at Θ (tilt) = 75.1 deg and (ellipticity) = 24.1 deg and with the global minimum at Θ (tilt) = 72.4 deg and (ellipticity) = −17.3 deg.14

Tables (4)

Tables Icon

Table 1 Roots and Intensities for Case 1

Tables Icon

Table 2 Optimal versus Standard Stokes Vectors for Case 1

Tables Icon

Table 3 Roots and Powers for Case 2

Tables Icon

Table 4 Optimal versus Standard Stokes Vectors for Case 2

Equations (49)

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E = [ E x E y ] = [ | E x | exp ( i δ x ) | E y | exp ( i δ y ) ] = { A ( t ) x ̂ + B ( t ) exp [ i ϕ ( t ) ] ŷ } exp ( i ω t ) ,
[ J ] = [ E x * E x E x * E y E y E x * E y E y * ] ,
lim T [ 1 2 T T + T ( ) d t ] .
p [ 1 4 ( det [ J ] ) ( J x x + J y y ) 2 ] 1 / 2 .
S [ S 0 S 1 S 2 S 3 ] [ J x x + J y y J x x J y y J x y + J y x i ( J x y J y x ) ]
p = ( S 1 2 + S 2 2 + S 3 2 ) 1 / 2 S 0 .
R = [ M ] T .
T 0 = ( T 1 2 + T 2 2 + T 3 2 ) 1 / 2
R = P + U ,
P = [ ( R 1 2 + R 2 2 + R 3 2 ) 1 / 2 R 1 R 2 R 3 ] , U = [ R 0 ( R 1 2 + R 2 2 + R 3 2 ) 1 / 2 0 0 0 ] .
p = ( R 1 2 + R 2 2 + R 3 2 ) 1 / 2 / R 0 .
Θ = ½ tan 1 { R 2 / R 1 } , π / 2 Θ π / 2 ,
= ½ sin 1 { R 3 / ( R 1 2 + R 2 2 + R 3 2 ) 1 / 2 } , π / 4 ε π / 4 ,
χ = E y E x = | E y | | E x | exp i ( δ y δ x ) = tan Θ + i tan 1 i tan Θ tan ,
h = E s *
[ 1 0 0 0 ] = ½ [ 1 0 0 + 1 ] LHC + ½ [ 1 0 0 1 ] RHC ,
R = P + U = [ p R 0 R 1 R 2 R 3 ] + [ ( 1 p ) R 0 0 0 0 ] ,
P = p R 0 + 1 / 2 ( 1 p ) R 0 = 1 / 2 R 0 ( 1 + p ) = 1 / 2 R 0 [ 1 + ( R 1 2 + R 2 2 + R 3 2 ) 1 / 2 R 0 ] ,
[ R 0 R ] = [ M 00 ñ T m [ M ] ] [ T 0 T ] ,
ñ [ M 01 M 03 ] , m [ M 10 M 30 ] , [ M ] [ M 11 M 13 M 31 M 33 ] .
R 0 = M 00 + ( ñ T , T ) ,
R = m + [ M ] T .
R [ M ] T , T = 1 ;
R 2 = ( R T , R ) = ( T T , [ G ] T ) ,
[ G ] T [ M ] [ M ] .
[ G ] T = λ T ,
det { [ G ] λ [ Ĩ ] } = 0 ,
R m + β m ;
R 2 ( R T , R ) = ( 1 + β ) 2 m 2 ,
R 2 = m 2 + 2 ( m T , [ M ] T ) + ( T T , [ G ] T ) , T = 1 ,
F = i j a i j x i x j + 2 b i x i + c , i | x i | 2 = 1 ,
a i j [ G ] , b i T m [ M ] , c m 2 , x i T .
F = i λ i y i 2 + 2 b i y i + c , b i j O j i b j , i , j = 1 , 2 , 3 ,
y i = j O i j x j x i = j O j i Y j .
( λ i μ ) y i = b i ,
y i = ( b i λ i μ ) .
i = 1 3 ( b i λ i μ ) 2 = 1 ,
| M i j | | M 11 | < | M 22 | | M 33 | < | M 00 | < 1 , i , j = 0 , 1 , 2 , 3 , i j
0 < M 00 ( i M 0 i 2 ) 1 / 2 < M 00 + ( i M 0 i 2 ) 1 / 2 < 1 , i = 1 , 2 , 3 ,
[ M ] = [ 0.7599 0.0623 0.0295 0.1185 0.0573 0.4687 0.1811 0.1863 0.0384 0.1714 0.5394 0.0282 0.1240 0.2168 0.0120 0.6608 ] .
b i T = m T [ M ] = [ 0.0603 0.0296 0.0937 ] .
a i j = [ G ] = [ 0.2961 0.1747 0.2354 0.1747 0.3239 0.0410 0.2354 0.0410 0.4722 ] .
λ 1 = 0.0683 , λ 2 = 0.3358 , λ 3 = 0.6880 ,
[ O ] = [ 0.7827 0.1872 0.5936 0.4696 0.8036 0.3656 0.4086 0.5649 0.7169 ] .
( 1.0000 ) μ 6 + ( 2.1842 ) μ 5 + ( 1.7814 ) μ 4 + ( 0.6780 ) μ 3 + ( 0.1221 ) μ 2 + ( 0.0092 ) μ 1 + ( 0.0002 ) μ 0 = 0.
[ M ] = [ 0.1800 0.0400 0.0100 0.0050 0.2900 0.1250 0.3100 0.0150 0.0200 0.0250 0.1600 0.3500 0.0150 0.1000 0.0800 0.1950 ] ,
λ 1 = 0.0088 , λ 2 = 0.0771 , λ 3 = 0.2292.
( 1.0000 ) μ 6 + ( 0.6302 ) μ 5 + ( 0.1308 ) μ 4 + ( 0.0100 ) μ 3 + ( 0.0002 ) μ 2 + ( 0.0000 ) μ 1 + ( 0.0000 ) μ 0 = 0 ,
A 0 = 1 , A 1 = 2 ( λ 1 + λ 2 + λ 3 ) , A 2 = ( λ 1 + λ 2 + λ 3 ) 2 + 2 ( λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) ( B 1 2 + B 2 2 + B 3 2 ) , A 3 = 2 [ λ 1 λ 2 λ 3 + ( λ 1 + λ 2 + λ 3 ) ( λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) ] + 2 [ B 1 2 ( λ 2 + λ 3 ) + B 2 2 ( λ 1 + λ 3 ) + B 3 2 ( λ 1 + λ 2 ) ] , A 4 = ( λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) 2 + 2 [ ( λ 1 λ 2 λ 3 ) ( λ 1 + λ 2 + λ 3 ) ] [ B 1 2 ( λ 2 + λ 3 ) 2 + B 2 2 ( λ 1 + λ 3 ) 2 + B 3 2 ( λ 1 + λ 2 ) 2 ] 2 [ B 1 2 ( λ 2 λ 3 ) + B 2 2 ( λ 1 λ 3 ) + B 3 2 ( λ 1 λ 2 ) ] , A 5 = 2 [ ( λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 ) ( λ 1 λ 2 λ 3 ) ] + 2 [ B 1 2 ( λ 2 λ 3 ) ( λ 2 + λ 3 ) + B 2 2 ( λ 1 λ 3 ) ( λ 1 + λ 3 ) + B 3 2 ( λ 1 λ 2 ) ( λ 1 + λ 2 ) ] , A 6 = ( λ 1 λ 2 λ 3 ) 2 [ B 1 2 ( λ 2 λ 3 ) 2 + B 2 2 ( λ 1 λ 3 ) 2 + B 3 2 ( λ 1 λ 2 ) 2 ] .

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