Abstract

The propagation characteristics of a spatially localized electromagnetic wave produced by a planar current source of different states of spatial coherence are analyzed by the use of a Gaussian Schell-model source. A linearly polarized fundamental electromagnetic Gaussian wave with the electric field perpendicular to the direction of propagation is treated. The effects of the degree of coherence of the source distribution on the radiation intensity distribution and the total radiated power are determined.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. H. Carter and M. Bertolotti, “An analysis of the far-field coherence and radiation intensity of light scattered from liquid crystals,” J. Opt. Soc. Am. 68, 329–333 (1978).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1984).
  3. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  5. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).
  6. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  7. S. R. Seshadri, “Dynamics of the linearly polarized fundamental Gaussian light wave,” J. Opt. Soc. Am. A 24, 482–492 (2007).
    [CrossRef]
  8. A. C. Schell, “The multiple plate antenna,” doctoral dissertation (Massachusetts Institute of Technology, 1961), Sect. 7.5.
  9. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
    [CrossRef]
  10. A. Burvall, A. Smith, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26, 1721–1729 (2009).
    [CrossRef]
  11. S. R. Seshadri, “Full-wave generalizations of the fundamental Gaussian beam,” J. Opt. Soc. Am. A 26, 2515–2520 (2009).
    [CrossRef]
  12. M. W. Kowarz and E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
    [CrossRef]

2009 (2)

2007 (2)

S. R. Seshadri, “Dynamics of the linearly polarized fundamental Gaussian light wave,” J. Opt. Soc. Am. A 24, 482–492 (2007).
[CrossRef]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

1999 (1)

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

1993 (1)

1984 (1)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1984).

1982 (1)

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).

1978 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

1961 (1)

A. C. Schell, “The multiple plate antenna,” doctoral dissertation (Massachusetts Institute of Technology, 1961), Sect. 7.5.

Bertolotti, M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1984).

Burvall, A.

Carter, W. H.

Dainty, C.

Kowarz, M. W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

Marathay, A. S.

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

A. C. Schell, “The multiple plate antenna,” doctoral dissertation (Massachusetts Institute of Technology, 1961), Sect. 7.5.

Seshadri, S. R.

Smith, A.

Wolf, E.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

M. W. Kowarz and E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1984).

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Other (5)

A. C. Schell, “The multiple plate antenna,” doctoral dissertation (Massachusetts Institute of Technology, 1961), Sect. 7.5.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1984).

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, 1982).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge Univ. Press, 2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Radiation intensity Φ ( θ , φ ) as a function of θ for 0 ° < θ < 90 ° , φ = 90 ° , k w 0 = 1.563 , and (a) w σ = 1 , (b) w σ = 0.9 , and (c) w σ = 0.8 . The normalization is such that the power carried by the fully coherent ( w σ = 1 ) wave in the paraxial approximation in the + z direction is given by P f 0 + = 1 W .

Fig. 2
Fig. 2

Radiation intensity Φ ( θ , φ ) as a function of θ for 0 ° < θ < 90 ° . The physical parameters are the same as those for Fig. 1 , except that k w 0 = 2.980 .

Fig. 3
Fig. 3

Total radiated power P + transported in the + z direction as a function of k w 0 for 1 < k w 0 < 5 . Other parameters are the same as in Fig. 1.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) A x ( x , y , z ) = J x ( x , y , z = 0 ) δ ( z ) .
( 2 x 2 + 2 y 2 + 2 z 2 + k 2 ) G ( x , y , z ) = δ ( x ) δ ( y ) δ ( z ) .
A x ( x , y , z ) = d x 1 d y 1 J x ( x 1 , y 1 , z 1 = 0 ) G ( x x 1 , y y 1 , z ) .
E x ( x , y , z ) = i k ( 1 + 1 k 2 2 x 2 ) A x ( x , y , z ) ,
H y ( x , y , z ) = A x ( x , y , z ) z .
E x ( x , y , z ) = d x 1 d y 1 J x ( x 1 , y 1 , z 1 = 0 ) × i k ( 1 + 1 k 2 2 x 2 ) G ( x x 1 , y y 1 , z ) .
H y ( x , y , z ) = d x 2 d y 2 J x ( x 2 , y 2 , z 2 = 0 ) × z G ( x x 2 , y y 2 , z ) .
S z ( r , t ) = c 2 Re d x 1 d y 1 d x 2 d y 2 J x ( x 1 , y 1 , z 1 = 0 ) J x * ( x 2 , y 2 , z 2 = 0 ) i k ( 1 + 1 k 2 2 x 2 ) G ( x x 1 , y y 1 , z ) z G * ( x x 2 , y y 2 , z ) ,
S z ( r ) = c 2 Re d x 1 d y 1 d x 2 d y 2 C 0 ( x 1 , y 1 , x 2 , y 2 ) × i k ( 1 + 1 k 2 2 x 2 ) G ( x x 1 , y y 1 , z ) × z G * ( x x 2 , y y 2 , z ) ,
C 0 ( x 1 , y 1 , x 2 , y 2 ) = J x ( x 1 , y 1 , z 1 = 0 ) J x * ( x 2 , y 2 , z 2 = 0 ) .
C 0 ( x 1 , y 1 , x 2 , y 2 ) = N 2 exp ( x 1 2 + y 1 2 w 0 2 ) exp ( x 2 2 + y 2 2 w 0 2 ) × g ( x 1 , y 1 , x 2 , y 2 ) ,
g ( x 1 , y 1 , x 2 , y 2 ) = exp { 1 σ g 2 [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] } ,
u c = 1 2 ( u 1 + u 2 )     and u d = u 1 u 2 for u = x , y .
W 0 ( x d , y d ; x c , y c ) = C 0 ( x c + 1 2 x d , y c + 1 2 y d , x c 1 2 x d , y c 1 2 y d ) = N 2 exp [ 2 ( x c 2 + y c 2 ) w 0 2 ] exp [ x d 2 + y d 2 σ t 2 ] ,
1 σ t 2 = 1 2 w 0 2 + 1 σ g 2 .
G ( x , y , z ) = d p x d p y exp [ i 2 π ( p x x + p y y ) ] G ¯ ( p x , p y , z )
δ ( x ) δ ( y ) = d p x d p y exp [ i 2 π ( p x x + p y y ) ] ,
( 2 z 2 + ζ 2 ) G ¯ ( p x , p y , z ) = δ ( z ) ,
ζ = [ k 2 4 π 2 ( p x 2 + p y 2 ) ] 1 2 .
G ¯ ( p x , p y , z ) = i 2 ζ exp ( i ζ | z | ) .
G ( x , y , z ) = i 2 d p x d p y exp [ i 2 π ( p x x + p y y ) ] ζ 1 exp ( i ζ | z | ) .
S z ( r ) = c k 8 Re d x 1 d y 1 × d x 2 d y 2 d p x 1 d p y 1 d p x 2 d p y 2 × W 0 ( x d , y d ; x c , y c ) ( 1 4 π 2 p x 1 2 k 2 ) × exp { i 2 π [ p x 1 ( x x 1 ) + p y 1 ( y y 1 ) ] } ζ 1 1 exp ( i ζ 1 z ) × exp { i 2 π [ p x 2 ( x x 2 ) + p y 2 ( y y 2 ) ] } exp ( i ζ 2 * z ) ,
P + = d x d y S z ( r ) = c k 8 Re d x d y d x 1 d y 1 d x 2 d y 2 d p x 1 d p y 1 d p x 2 d p y 2 × W 0 ( x d , y d ; x c , y c ) ( 1 4 π 2 p x 1 2 k 2 ) exp [ i 2 π ( p x 1 x 1 + p y 1 y 1 ) ] ζ 1 1 exp ( i ζ 1 z ) exp [ i 2 π ( p x 2 x 2 + p y 2 y 2 ) ] exp ( i ζ 2 * z ) × exp { i 2 π [ x ( p x 1 p x 2 ) + y ( p y 1 p y 2 ) ] } .
P + = c k 8 Re d x 1 d y 1 d x 2 d y 2 d p x 1 d p y 1 × W 0 ( x d , y d ; x c , y c ) ( 1 4 π 2 p x 1 2 k 2 ) × exp [ i 2 π ( p x 1 x d + p y 1 y d ) ] ζ 1 1 exp [ i z ( ζ 1 ζ 1 * ) ] .
I x 1 x 2 = d x c exp ( 2 x c 2 w 0 2 ) × d x d exp ( x d 2 σ t 2 ) exp ( i 2 π p x 1 x d ) .
I x 1 x 2 = π 2 w 0 σ t exp ( π 2 σ t 2 p x 1 2 ) .
I y 1 y 2 = π 2 w 0 σ t exp ( π 2 σ t 2 p y 1 2 ) .
P + = N 2 c k 16 π 2 w 0 2 σ t 2 Re d p x 1 d p y 1 × ( 1 4 π 2 p x 1 2 k 2 ) exp [ π 2 σ t 2 ( p x 1 2 + p y 1 2 ) ] × ζ 1 1 exp [ i z ( ζ 1 ζ 1 * ) ] .
2 π p x 1 = p cos φ and 2 π p y 1 = p sin φ
P + = N 2 c k 64 w 0 2 σ t 2 Re 0 d p p 0 2 π d φ ( 1 p 2 k 2 cos 2 φ ) × exp ( 1 4 σ t 2 p 2 ) ( k 2 p 2 ) 1 2 × exp { i z [ ( k 2 p 2 ) 1 2 ( k 2 p 2 ) 1 2 * ] } .
P + = N 2 c k 64 w 0 2 σ t 2 0 k d p p 0 2 π d φ ( 1 p 2 k 2 cos 2 φ ) × exp ( 1 4 σ t 2 p 2 ) ( k 2 p 2 ) 1 2 .
P f + = N 2 c k 32 w 0 4 0 k d p p 0 2 π d φ ( 1 p 2 k 2 cos 2 φ ) × exp ( 1 2 w 0 2 p 2 ) ( k 2 p 2 ) 1 2 ,
P + = k σ t 2 4 π 0 k d p p 0 2 π d φ ( 1 p 2 k 2 cos 2 φ ) × exp ( 1 4 σ t 2 p 2 ) ( k 2 p 2 ) 1 2 .
σ t 2 = 2 w 0 2 w σ 2 ,   where w σ 2 = ( 1 + 2 w 0 2 σ g 2 ) 1
P + = 0 π 2 d θ sin θ 0 2 π d φ Φ ( θ , φ ) ,
Φ ( θ , φ ) = k 2 w 0 2 w σ 2 2 π ( 1 sin 2 θ cos 2 φ ) exp ( 1 2 k 2 w 0 2 w σ 2 sin 2 θ ) .

Metrics