Abstract

Using the coherence matrices derived in Part I of this investigation, a systematic study is made of the structure of the far field, on the basis of several different theories. First the case of a completely polarized incident field is considered and the distribution of the energy density and the state of polarization are examined. Then the case of diffraction of an unpolarized field is studied, particular attention being paid to the degree of polarization of the diffracted field. Analytical as well as some numerical results are presented. The predictions of the three theories are compared with each other and with those of the classical scalar theory of Huygens, Fresnel, and Kirchhoff.

© 1966 Optical Society of America

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References

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  1. B. Karczewski and E. Wolf, J. Opt. Soc. Am. 56, 1207 (1966).
    [CrossRef]
  2. E. Wolf, Nuovo Cimento 13, 1180 (1959).
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.
  4. In Refs. 1 and 2 the elements of the coherence matrix have been defined in terms of a time average rather than in terms of an ensemble average as done in the present paper. However, since we are dealing with a stationary field, which we also assumed to be ergodic, the two averages are the same.

1966 (1)

1959 (1)

E. Wolf, Nuovo Cimento 13, 1180 (1959).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

Karczewski, B.

Wolf, E.

B. Karczewski and E. Wolf, J. Opt. Soc. Am. 56, 1207 (1966).
[CrossRef]

E. Wolf, Nuovo Cimento 13, 1180 (1959).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

J. Opt. Soc. Am. (1)

Nuovo Cimento (1)

E. Wolf, Nuovo Cimento 13, 1180 (1959).

Other (2)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed., Chap. 8.

In Refs. 1 and 2 the elements of the coherence matrix have been defined in terms of a time average rather than in terms of an ensemble average as done in the present paper. However, since we are dealing with a stationary field, which we also assumed to be ergodic, the two averages are the same.

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Figures (4)

F. 1
F. 1

Behavior of the angle χ, which specifies the orientation of the vibrational ellipse, as function of the angle θ of diffraction. of the vibrational ellipse, as function of the angle θ of diffrac-Normal incidence (K = 0,0,1). |Ay|/|Ay| = 2, δ = 60°,k = [0,ky,kz].

F. 2
F. 2

The behavior of the modulation factor M as function of the angle of diffraction θ. Normal incidence. Symbol ∥(⊥) denotes the case when the electric vector of the incident field vibrates in the direction parallel (perpendicular) to the plane specified by the direction of incidence and observation.

F. 3
F. 3

Enlargement of central portion of Fig. 2.

F. 4
F. 4

Degree of polarization of the diffracted field as function of the angle of diffraction θ, for the case of normally incident unpolarized plane quasimonochromatic field.

Equations (39)

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J l m = J m l * = | E l | | E m | e i ϕ J l l = | E l | 2 , J m m = | E m | 2 , }
J l m / ( J l l ) 1 2 ( J m m ) 1 2 = e i ϕ ;
cos ϕ = 1 2 ( J l m + J m l ) / ( J l l ) 1 2 ( J m m ) 1 2 sin ϕ = ( 1 / 2 i ) ( J l m J m l ) / ( J l l ) 1 2 ( J m m ) 1 2 . }
tan 2 χ = ( J l m + J m l ) / ( J l l J m m ) .
tan ϕ ( m ) = a 1 ( m ) | A L | sin δ b 1 ( m ) | A M | + a 1 ( m ) | A L | cos δ ,
tan 2 χ ( m ) = 2 b 1 ( m ) b 2 ( m ) | A M | 2 + 2 a 1 ( m ) b 2 ( m ) | A L | | A M | cos δ ( a 1 ( m ) ) 2 | A 2 | 2 + [ ( b 1 ( m ) ) 2 ( b 2 ( m ) ) 2 ] | A M | 2 + 2 a 1 ( m ) b 1 ( m ) | A L | | A M | cos δ .
tan ϕ ( e ) = b 2 ( e ) | A M | sin δ a 2 ( e ) | A L | + b 2 ( e ) | A M | cos δ ,
ϕ ( e ) = M π ( M = 0 , ± 1 , ± 2 , ) .
tan 2 χ ( e ) = 2 a 1 ( e ) [ a 2 ( e ) | A L | 2 + b 2 ( e ) | A L | | A M | cos δ ] [ ( a 1 ( e ) ) 2 ( a 2 ( e ) ) 2 ] | A L | 2 ( b 2 ( e ) ) 2 | A M | 2 2 a 2 ( e ) b 2 ( e ) | A L | | A M | cos δ .
tan ϕ ( e , m ) = ( a 1 ( e , m ) b 2 ( e , m ) a 2 ( e , m ) b 1 ( e , m ) ) | A L | | A M | sin δ a 1 ( e , m ) a 2 ( e , m ) | A L | 2 + b 1 ( e , m ) b 2 ( e , m ) | A M | 2 + ( a 1 ( e , m ) b 2 ( e , m ) + a 2 ( e , m ) b 1 ( e , m ) ) | A L | | A M | cos δ ,
ϕ ( e , m ) = M π , ( M = 0 , ± 1 , ± 2 , ) .
ϕ ( e , m ) = N π / 2 , ( N = ± 1 , ± 3 ) .
tan 2 χ ( e , m ) = 2 [ a 1 ( e , m ) a 2 ( e , m ) | A L | 2 + b 1 ( e , m ) b 2 ( e , m ) | A M | 2 + ( a 1 ( e , m ) b 2 ( e , m ) + a 2 ( e , m ) b 1 ( e , m ) ) | A L | | A M | cos δ ] [ ( a 1 ( e , m ) ) 2 ( a 2 ( e , m ) ) 2 ] | A L | 2 + [ ( b 1 ( e , m ) ) 2 ( b 2 ( e , m ) ) 2 ] | A M | 2 + 2 ( a 1 ( e , m ) b 1 ( e , m ) a 2 ( e , m ) b 2 ( e , m ) ) | A L | | A M | cos δ .
W = ( 1 / 16 π ) ( J l l + J m m ) .
W ( m ) = ( 1 / 4 π ) | F | 2 { ( a L ( m ) ) 2 | A L | 2 + [ ( b 1 ( m ) ) 2 + ( b 2 ( m ) ) 2 ] | A M | 2 + 2 a 1 ( m ) b 1 ( m ) | A L | | A M | cos δ } ,
W ( e ) = ( 1 / 4 π ) | F | 2 { [ ( a 1 ( e ) ) 2 ( a 2 ( e ) ) 2 ] | A L | 2 + ( b 2 ( e ) ) 2 | A M | 2 + 2 a 2 ( e ) b 2 ( e ) | A L | | A M | cos δ } ,
W ( e , m ) = ( 1 / 4 π ) | F | 2 { [ ( a 1 ( e , m ) ) 2 + ( a 2 ( e , m ) ) 2 ] | A L | 2 + [ ( b 1 ( e , m ) ) 2 + ( b 2 ( e , m ) ) 2 ] | A M | 2 + 2 ( a 1 ( e , m ) b 1 ( e , m ) + a 2 ( e , m ) b 2 ( e , m ) ) × | A L | | A M | cos δ } ,
F = C a exp [ i ( 2 π / λ ) ( K k ) · r ] d S .
W ( m ) = ( 1 / 4 π ) | F | 2 | A L | 2 ,
W ( e ) = ( 1 / 4 π ) | F | 2 | A L | 2 cos 2 θ ,
W ( e , m ) = ( 1 / 4 π ) | F | 2 | A L | 2 cos 4 ( θ / 2 ) ,
cos θ = k z .
W ( m ) = ( 1 / 4 π ) | F | 2 | A M | 2 cos 2 θ ,
W ( e ) = ( 1 / 4 π ) | F | 2 | A M | 2 ,
W ( e , m ) = ( 1 / 4 π ) | F | 2 | A M | 2 cos 4 ( θ / 2 ) .
W ( F ) = ( 1 / 4 π ) | F | 2 | A | ,
W ( α ) = M ( α ) W ( F ) ,
A L = A cos α , A M = A sin α .
J ¯ ( k ) = [ E l ( k ) E l * ( k ) E l ( k ) E m * ( k ) E m ( k ) E l * ( k ) E m ( k ) E m * ( k ) ] .
cos 2 α = sin 2 α = 1 2 , sin α cos α = 0 .
J ¯ l l ( m ) = 2 | F | 2 | A | 2 [ ( a 1 ( m ) ) 2 + ( b 1 ( m ) ) 2 ] , J ¯ l m ( m ) = 2 | F | 2 | A | 2 b 1 ( m ) b 2 ( m ) , J ¯ m l ( m ) = ( J ¯ l m ( m ) ) * , J ¯ m m ( m ) = 2 | F | 2 | A | 2 ( b 2 ( m ) ) 2 ; }
J ¯ l l ( e ) = 2 | F | 2 | A | 2 ( a 1 ( e ) ) 2 , J ¯ l m ( e ) = 2 | F | 2 | A | 2 a 1 ( e ) a 2 ( e ) , J ¯ m l ( e ) = ( J ¯ l m ( e ) ) * , J ¯ m m ( e ) = 2 | F | 2 | A | 2 [ ( a 2 ( e ) ) 2 + ( b 2 ( e ) ) 2 ] ; }
J ¯ l l ( e , m ) = 2 | F | 2 | A | 2 [ ( a 1 ( e , m ) ) 2 + ( b 1 ( e , m ) ) 2 ] , J ¯ l m ( e , m ) = 2 | F | 2 | A | 2 × [ a 1 ( e , m ) a 2 ( e , m ) + b 1 ( e , m ) b 2 ( e , m ) ] , J ¯ m l ( e , m ) = ( J ¯ l m ( e , m ) ) * , J ¯ m m ( e , m ) = 2 | F | 2 | A | 2 [ ( a 2 ( e , m ) ) 2 + ( b 2 ( e , m ) ) 2 ] . }
W = ( 1 / 16 π ) ( J ¯ l l + J ¯ m m )
W ( m ) = 2 | F | 2 | A | 2 × [ ( a 1 ( m ) ) 2 + ( b 1 ( m ) ) 2 + ( b 2 ( m ) ) 2 ] , W ( e ) = 2 | F | 2 | A | 2 [ ( a 1 ( e ) ) 2 + ( a 2 ( e ) ) 2 + ( b 2 ( e ) ) 2 ] , W ( e , m ) = 2 | F | 2 | A | 2 [ ( a 1 ( e , m ) ) 2 + ( a 2 ( e , m ) ) 2 + ( b 1 ( e , m ) ) 2 + ( b 2 ( e , m ) ) 2 ] . }
P = ( 1 4 | J ¯ | / ( J ¯ l l + J ¯ m m ) 2 ) 1 2 ,
P ( m ) = ( 1 4 k z 2 K z 2 [ K z 2 + K z 2 + ( k y K x k x K y ) 2 ] 2 ) 1 2 .
P ( e ) = P ( m ) .
P ( e , m ) = 0 .