Abstract

The independent-elementary-source model recently developed for planar sources [Opt. Express 14, 1376 (2006)] is extended to volume source distributions. It is shown that the far-field radiation pattern is independent of the three-dimensional (3D) distribution of the coherent elementary sources, but the absolute value of the complex degree of spectral coherence is determined by the 3D Fourier transform of the weight function of the elementary fields. Some methods to determine an ‘effective’ three-dimensional source distribution with partial transverse and longitudinal spatial coherence properties are outlined. Especially in its longitudinal extension, this effective source volume can be very different from the primary emitting volume of the source. The application of the model to efficient numerical propagation of partially coherent fields is discussed.

© 2008 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  2. P. Vahimaa and J. Turunen, "Finite-elementary source model for partially coherent radiation," Opt. Express 14,1376 (2006).
    [CrossRef] [PubMed]
  3. F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978).
    [CrossRef]
  4. F. Gori, "Directionality and spatial coherence," Opt. Acta 27, 1025-1034 (1980).
    [CrossRef]
  5. F. Gori and M. Santarsiero, "Devising genuine correlation functions," Opt. Lett. 32, 3531-3133 (2007).
    [CrossRef] [PubMed]
  6. M. Peeters, G. Verschaffelt, H. Thienpont, S. K. Mandre, P. Vahimaa, and J. Turunen, "Propagation of spatially partially coherent emission from a vertical-cavity surface-emitting laser," Opt. Express 13, 9337-9345 (2005).
    [CrossRef] [PubMed]
  7. P. Vahimaa and J. Turunen, "Independent-elementary-pulse representation for non-stationary fields," Opt. Express 14, 5007-5012 (2006).
    [CrossRef] [PubMed]
  8. A. T. Friberg, H. Lajunen, and V. Torres-Company, "Spectral elementary-coherence-function representation for partially coherent light pulses," Opt. Express 15, 5160-5165 (2007). http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-8-5160
    [CrossRef] [PubMed]
  9. V. Torres-Company, G. Minguez-Vega, J. Lancis, and A. T. Friberg, "Controllable generation of partially coherent light pulses with direct space-to-time pulse shaper," Opt. Lett. 32, 1608-1010 (2007).
    [CrossRef] [PubMed]
  10. A. T. Friberg and R. J. Sudol, "The spatial coherence properties of gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
    [CrossRef]
  11. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168- 1177 (1970).
    [CrossRef]
  12. J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

2007

2006

2005

2000

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

1983

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

1980

F. Gori, "Directionality and spatial coherence," Opt. Acta 27, 1025-1034 (1980).
[CrossRef]

1978

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978).
[CrossRef]

1970

Collins, S. A.

Friberg, A. T.

Gori, F.

F. Gori and M. Santarsiero, "Devising genuine correlation functions," Opt. Lett. 32, 3531-3133 (2007).
[CrossRef] [PubMed]

F. Gori, "Directionality and spatial coherence," Opt. Acta 27, 1025-1034 (1980).
[CrossRef]

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978).
[CrossRef]

Kaivola, M.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Kajava, T.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Kuittinen, M.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Laakkonen, P.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Lajunen, H.

Mandre, S. K.

Pääkkönen, P.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Palma, C.

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978).
[CrossRef]

Peeters, M.

Santarsiero, M.

Simonen, J.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

Sudol, R. J.

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

Thienpont, H.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Verschaffelt, G.

J. Mod. Opt.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, "Diffractive shaping of excimer laser beams," J. Mod. Opt. 47, 2467-2475 (2000).

J. Opt. Soc. Am.

Opt. Acta

A. T. Friberg and R. J. Sudol, "The spatial coherence properties of gaussian Schell-model beams," Opt. Acta 30, 1075-1097 (1983).
[CrossRef]

F. Gori, "Directionality and spatial coherence," Opt. Acta 27, 1025-1034 (1980).
[CrossRef]

Opt. Commun.

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional fields," Opt. Commun. 27, 185-187 (1978).
[CrossRef]

Opt. Express

Opt. Lett.

Other

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

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Figures (5)

Fig. 1.
Fig. 1.

Procedure of propagating the elementary-mode superposition from the source volume VS into the target volume VT . Points P′=(x′,y′,z′) within VS are the centers of origin of the elementary fields f(x,y), which are propagated over the set of distances indicated by the arrows along the z-axis to map VS onto VT .

Fig. 2.
Fig. 2.

Longitudinally extended volume source consisting of a superposition of axial Gaussian elementary sources in the interval -L<z<0. The distributions of normalized (to the value at z=0) axial intensity are shown for different values L, assuming w 0=100λ.

Fig. 3.
Fig. 3.

Source models as in Fig. 2, but distributions of longitudinal axial spatial coherence between points z 1=0 and and arbitrary plane z 2=z are shown.

Fig. 4.
Fig. 4.

Source models as in Fig. 2, but normalized (to the value at the origin) transverse intensity distributions at the exit plane z=0 of the source volume are shown.

Fig. 5.
Fig. 5.

Source models as in Fig. 2, but distributions of the absolute value of the complex degree of transverse spatial coherence at the end plane of the source volume z=0 are shown between points (x 1,y 1)=(0,0) and (x 2,y 2)=(x,0).

Equations (30)

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F ( k x , k y ) = 1 ( 2 π ) 2 f ( x , y ) exp [ i ( k x x + k y y ) ] d x d y ,
k z = { ( k 2 k x 2 k y 2 ) 1 2 if k x 2 + k y 2 k 2 i ( k x 2 + k y 2 k 2 ) 1 2 if k x 2 + k y 2 > k 2 ,
a ( k x , k y ; x , y , z ) = c ( x , y , z ) F ( k x , k y ) exp [ i ( k x x + k y y ) ] exp ( i k z z ) ,
A ( k x 1 , k y 1 , k x 2 , k y 2 ) = a * ( k x 1 , k y 1 ; x 1 , y 1 , z 1 ) a ( k x 2 , k y 2 ; x 2 , y 2 , z 2 ) d x 1 d y 1 d z 1 d x 2 d y 2 d z 2
c * ( x 1 , y 1 , z 1 ) c ( x 2 , y 2 , z 2 ) = p ( x 1 , y 1 , z 1 ) δ ( x 1 x 2 , y 1 y 2 , z 1 z 2 ) ,
A ( k x 1 , k y 1 , k x 2 , k y 2 ) = F * ( k x 1 , k y 1 ) F ( k x 2 , k y 2 ) P ( k x 2 k x 1 , k y 2 k y 1 , k z 2 k z 1 * ) ,
P ( k x , k y , k z ) = 1 ( 2 π ) 3 p ( x , y , z ) exp [ i ( k x x + k y y + k z z ) ] d x d y d z
W ( ) ( r 1 s x 1 , r 1 s y 1 , r 2 s x 2 , r 2 s y 2 ) = ( 2 π k ) 2 s z 1 s z 2 F * ( k s x 1 , k s y 1 ) F ( k s x 2 , k s y 2 )
× P ( k s x 2 k s x 1 , k s y 2 k s y 1 , k s z 2 k s z 1 ) exp [ i k ( r 2 r 1 ) ] r 1 r 2 .
J ( s x , s y ) = r 2 W ( ) ( k s x , k s y , k s x , k s y ) = ( 2 π k ) 2 s z 2 P ( 0,0,0 ) F ( k s x , k s y ) 2
μ ( ) ( r 1 s x 1 , r 1 s y 1 , r 2 s x 2 , r 2 s y 2 ) = P ( k s x 2 k s x 1 , k s y 2 k s y 1 , k s z 2 k s z 1 ) P ( 0,0,0 )
× exp { i [ k ( r 2 r 1 ) + arg F ( k s x 2 , k s y 2 ) arg F ( k s x 1 , k s y 1 ) ] } .
W ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) = F * ( k x 1 , k y 1 ) F ( k x 2 , k y 2 )
× P ( k x 2 k x 1 , k y 2 k y 1 , k z 2 k z 1 * )
× exp [ i ( x 1 k x 1 x 2 k x 2 + y 1 k y 1 y 2 k y 2 + z 1 k z 1 * z 2 k z 2 ) ] d k x 1 d k x 2 d k y 1 d k y 2 ,
W ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) = p ( x , y , z )
× f * ( x 1 x , y 1 y , z 1 z ) f ( x 2 x , y 2 y , z 2 z ) d x d y d z ,
V ( s x , s y ) = μ ( ) ( 0,0 , r s x , r s y ) = P ( k s x , k s y , k s z k ) P ( 0,0,0 )
= p ( x , y , z ) exp { i k [ s x x + s y y + ( s z 1 ) z ] } d x d y d z p ( x , y , z ) d x d y d z ,
p ( x , y , z ) = { p 0 if D 2 < x < D 2 , D 2 < y < D 2 , L < z < 0 0 otherwise
V ( s x , 0 ) = sinc ( π D λ sin θ ) sinc [ π L λ ( 1 cos θ ) ] ,
W ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ) = exp [ i k ( z 2 z 1 ) ] 0 p L ( z )
× [ 1 i ( z 1 z ) z R ] 1 [ 1 + i ( z 2 z ) z R ] 1
× exp { x 1 2 + y 1 2 w 0 2 [ 1 i ( z 1 z ) z R ] } exp { x 2 2 + y 2 2 w 0 2 [ 1 + i ( z 1 z ) z R ] } d z ,
0 p L ( z ) d z = 1 ,
μ ( x 1 , y 1 , z , x 2 , y 2 , z ) = W ( x 1 , y 1 , z , x 2 , y 2 , z ) [ I ( x 1 , y 1 , z ) I ( x 2 , y 1 , z ) ] 1 2 ,
μ ( 0 , 0 , z 1 , 0 , 0 , z 2 ) = W ( 0 , 0 , z 1 , 0 , 0 , z 2 ) [ I ( 0 , 0 , z 1 ) I ( 0 , 0 , z 2 ) ] 1 2 ,
W ( 0 , 0 , z 1 , 0 , 0 , z 2 ) = z R 2 exp ( i k Δ z ) L ( Δ z i 2 z R ) ( ln z 1 + L + i z R z 2 + L i z R ln z 1 + i z R z 2 i z R )
I ( 0 , 0 , z ) = z R L [ arctan ( z + L z R ) arctan ( z z R ) ] .
μ ( 0 , 0 , 0 , 0 , 0 , z ) = exp ( i k z ) ln ( 1 + i L z R ) ( L z R ) arctan ( L z R ) ,

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