Abstract

Evidence of the physical existence of the spatial coherence moiré is obtained by confronting numerical results with experimental results of spatially partial interference. Although it was performed for two particular cases, the results reveal a general behavior of the optical fields in any state of spatial coherence. Moreover, the study of the spatial coherence moiré deals with a new type of filtering, named filtering of classes of radiator pairs, which allows changing the power spectrum at the observation plane by modulating the complex degree of spatial coherence, without altering the power distribution at the aperture plane or introducing conventional spatial filters. This new procedure can optimize some technological applications of actual interest, as the beam shaping for instance.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Castañeda and J. García, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
    [CrossRef]
  2. R. Castañeda and J. García, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).
  3. R. Castañeda and J. García, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
    [CrossRef]
  4. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  5. R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
    [CrossRef]
  6. R. Castañeda and J. García-Sucerquia, "Radiometry and spatial coherence wavelets," Opt. Commun. 248, 147-165 (2005).
    [CrossRef]
  7. J. García and R. Castañeda, "Full retrieving of the complex degree of spatial coherence: theoretical analysis," Opt. Commun. 228, 9-19 (2003).
    [CrossRef]
  8. J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).
  9. R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
    [CrossRef]
  10. R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
    [CrossRef]

2007 (1)

R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
[CrossRef]

2005 (2)

R. Castañeda and J. García-Sucerquia, "Radiometry and spatial coherence wavelets," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
[CrossRef]

2004 (1)

R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
[CrossRef]

2003 (4)

J. García and R. Castañeda, "Full retrieving of the complex degree of spatial coherence: theoretical analysis," Opt. Commun. 228, 9-19 (2003).
[CrossRef]

R. Castañeda and J. García, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castañeda and J. García, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. García, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

1995 (1)

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

1978 (1)

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

Carrasquilla, J.

R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
[CrossRef]

Castaneda, R.

R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
[CrossRef]

Castañeda, R.

R. Castañeda and J. García-Sucerquia, "Radiometry and spatial coherence wavelets," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
[CrossRef]

J. García and R. Castañeda, "Full retrieving of the complex degree of spatial coherence: theoretical analysis," Opt. Commun. 228, 9-19 (2003).
[CrossRef]

R. Castañeda and J. García, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

R. Castañeda and J. García, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. García, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

García, J.

R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
[CrossRef]

J. García and R. Castañeda, "Full retrieving of the complex degree of spatial coherence: theoretical analysis," Opt. Commun. 228, 9-19 (2003).
[CrossRef]

R. Castañeda and J. García, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

R. Castañeda and J. García, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. García, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

García-Sucerquia, J.

R. Castañeda and J. García-Sucerquia, "Radiometry and spatial coherence wavelets," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

Herrera, J.

R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
[CrossRef]

Mandel, L.

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

Wolf, E.

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

J. Mod. Opt. (2)

R. Castañeda and J. García, "Spatial coherence wavelets," J. Mod. Opt. 50, 1259-1275 (2003).

R. Castañeda and J. García, "Spatial coherence wavelets: mathematical properties and physical features," J. Mod. Opt. 50, 2741-2753 (2003).
[CrossRef]

Opt. Commun. (6)

R. Castaneda, J. García, and J. Carrasquilla, "Quality descriptors of optical beams based on centred reduced moments III: spot moments-based method for laser characterization," Opt. Commun. 248, 509-519 (2005).
[CrossRef]

R. Castaneda, J. Carrasquilla, and J. Herrera, "Radiometric analysis of diffraction of quasi-homogeneous optical fields," Opt. Commun. 273, 8-20 (2007).
[CrossRef]

R. Castañeda, "Partially coherent imaging and spatial coherence wavelets," Opt. Commun. 230, 7-18 (2004).
[CrossRef]

R. Castañeda and J. García-Sucerquia, "Radiometry and spatial coherence wavelets," Opt. Commun. 248, 147-165 (2005).
[CrossRef]

J. García and R. Castañeda, "Full retrieving of the complex degree of spatial coherence: theoretical analysis," Opt. Commun. 228, 9-19 (2003).
[CrossRef]

R. Castañeda and J. García, "Classes of source pairs in interference and diffraction," Opt. Commun. 226, 45-55 (2003).
[CrossRef]

Other (2)

J. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, 1978).

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

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 (6)

Fig. 1
Fig. 1

Illustrating the concept of class of radiator pairs. Each circle within the aperture represents the support of the complex degree of spatial coherence centered at the position r A ( j ) . The radiator pairs of separation vectors r D ( j ) belong to the same class.

Fig. 2
Fig. 2

Illustrating the spatial coherence moiré.

Fig. 3
Fig. 3

Pinhole-masks used for analyzing the spatial coherence moiré and the map of classes of radiator pairs of an optical field, within controlled conditions of spatial coherence.

Fig. 4
Fig. 4

Numerically calculated Fraunhofer interference patterns and their corresponding maps of classes of radiator pairs (just below each interference pattern). Note that r ˜ D = x ˜ D i + y ˜ D j .

Fig. 5
Fig. 5

Experimental setup.

Fig. 6
Fig. 6

Experimental Fraunhofer interference patterns and their corresponding maps of classes of radiator pairs (just below each interference pattern) generated under similar conditions to those assumed for the numerical calculations. Images under the same letter in Figs. 4 and 6 are in correspondence.

Tables (1)

Tables Icon

Table 1 Measured Support Radii of the Gaussian Degree of Spatial Coherence R D ′, Using the Method in [9]

Equations (9)

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

S ( r A ; ω ) = ( 1 λ z ) 2 A W ( r A , r A ; ω ) d 2 r A .
W ( r A + r D 2 , r A r D 2 , r A ; ω ) = S ( r A , r A ; ω ) × exp ( i k z r D · r A ) ,
S ( r A , r A ; ω ) = λ z S ( r A ; ω ) | t ( r A ) | 2 + 2 A r D 0 | μ ( r A + r D 2 , r A r D 2 ; ω ) | × S ( r A + r D 2 ; ω ) | t ( r A + r D 2 ) | × S ( r A r D 2 ; ω ) | t ( r A r D 2 ) | × cos [ k z ( r A r A ) · r D + α 12 + ϕ 12 ] d 2 r D ,
S ( r A ; ω ) = 1 λ z S 0 ( ω ) A | t ( r A ) | 2 d 2 r A + 2 ( 1 λ z ) 2 × S 0 ( ω ) A A r D 0 | μ ( r A + r D 2 , r A r D 2 ; ω ) | × t ( r A + r D 2 ) t ( r A r D 2 ) × cos [ k z r A · r D α 12 ] d 2 r A d 2 r D ,
S ˜ ( r ˜ D ; ω ) = S 0 ( ω ) A | t ( r A ) | 2 d 2 r A δ ( r ˜ D ) + S 0 ( ω ) ( 1 λ z ) × A r ˜ D 0 | μ ( r A + r ˜ D 2 , r A r ˜ D 2 ; ω ) | × exp [ i α ( r A + r ˜ D 2 , r A r ˜ D 2 ; ω ) ] × t ( r A + r ˜ D 2 ) t ( r A r ˜ D 2 ) d 2 r A ,
S ( r A ; ω ) = 2 ( 1 λ z ) 2 S 0 ( ω ) { 2 + μ ( a ; ω ) cos ( k z a · r A ) + μ ( b ; ω ) cos ( k z b · r A ) + μ ( a + b + 2 c 2 ; ω ) × cos [ k z ( a + b + 2 c 2 ) · r A ] + μ ( a + b 2 c 2 ; ω ) × cos [ k z ( a + b 2 c 2 ) · r A ] + μ ( a b + 2 c 2 ; ω ) × cos [ k z ( a b + 2 c 2 ) · r A ] + μ ( a b 2 c 2 ; ω ) × cos [ k z ( a b 2 c 2 ) · r A ] } ,
S ( r A ; ω ) = 2 ( 1 λ z ) 2 S 0 ( ω ) { 2 + μ ( a ; ω ) cos ( k z a · r A ) + μ ( b ; ω ) cos ( k z b · r A ) + 2 μ ( a + b 2 ; ω ) × cos [ k z ( a + b 2 ) · r A ] + 2 μ ( a b 2 ; ω ) × cos [ k z ( a b 2 ) · r A ] } ,
S ˜ ( r ˜ D ; ω ) = S 0 ( ω ) { 4 δ ( r ˜ D ) + μ ( a ; ω ) [ δ ( r ˜ D + a ) + δ ( r ˜ D a ) ] + μ ( b ; ω ) [ δ ( r ˜ D + b ) + δ ( r ˜ D b ) ] + μ ( a + b + 2 c 2 ; ω ) [ δ ( r ˜ D + a + b + 2 c 2 ) + δ ( r ˜ D a + b + 2 c 2 ) ] + μ ( a + b 2 c 2 ; ω ) × [ δ ( r ˜ D + a + b 2 c 2 ) + δ ( r ˜ D a + b 2 c 2 ) ] + μ ( a b + 2 c 2 ; ω ) [ δ ( r ˜ D + a b + 2 c 2 ) + δ ( r ˜ D a b + 2 c 2 ) ] + μ ( a b 2 c 2 ; ω ) × [ δ ( r ˜ D + a b 2 c 2 ) + δ ( r ˜ D a b 2 c 2 ) ] } .
S ˜ ( r ˜ D ; ω ) = S 0 ( ω ) { 4 δ ( r ˜ D ) + μ ( a ; ω ) [ δ ( r ˜ D + a ) + δ ( r ˜ D a ) ] + μ ( b ; ω ) [ δ ( r ˜ D + b ) + δ ( r ˜ D b ) ] + 2 μ ( a + b 2 ; ω ) [ δ ( r ˜ D + a + b 2 ) + δ ( r ˜ D a + b 2 ) ] + 2 μ ( a b 2 ; ω ) × [ δ ( r ˜ D + a b 2 ) + δ ( r ˜ D a b 2 ) ] } ,

Metrics