Abstract

The phase-space representation of the Fresnel–Fraunhofer diffraction of optical fields in any state of spatial coherence is based on the marginal power spectrum carried by the spatial coherence wavelets. Its structure is analyzed in terms of the classes of source pairs and the spot of the field, which is treated as the hologram of the map of classes. Negative values of the marginal power spectrum are interpreted as negative energies. The influence of the aperture edge on diffraction is stated in terms of the distortion of the supports of the complex degree of spatial coherence near it. Experimental results are presented.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  2. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215-1224 (1981).
    [CrossRef]
  3. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227-1237 (1986).
    [CrossRef]
  4. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
    [CrossRef]
  5. R. Simon and N. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440-2463 (2000).
    [CrossRef]
  6. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  7. E. W. Marchand and E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 641219-1226 (1974).
    [CrossRef]
  8. R. Castañeda and J. García-Sucerquia, “Radiometry and spatial coherence wavelets,” Opt. Commun. 248, 147-165 (2005).
    [CrossRef]
  9. R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
    [CrossRef]
  10. R. Castañeda and J. García-Sucerquia, “Spatial coherence wavelets,” J. Mod. Opt. 50, 1259-1275 (2003).
  11. R. Castañeda and J. García-Sucerquia, “Spatial coherence wavelets: mathematical properties and physical features,” J. Mod. Opt. 50, 2741-2753 (2003).
    [CrossRef]
  12. R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 81-90 (2006).
    [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  14. R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
    [CrossRef]
  15. R. Castaneda, M. Usuga-Castaneda, and J. Herrera-Ramirez, “Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs,” Appl. Opt. 46, 5321-5328 (2007).
    [CrossRef] [PubMed]
  16. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).
  17. R. Castaneda, J. García, and J. Carrasquilla, “Quality descriptors of optical beams based on centered reduced moments. III: Spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
    [CrossRef]
  18. D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).
  19. R. Castaneda, J. Carrasquilla, and J. Herrera, “Radiometric analysis of diffraction of quasi-homogeneous optical fields,” Opt. Commun. 273, 8-20 (2007).
    [CrossRef]
  20. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series, Vol. MS 69 (SPIE, 1993).

2007

R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[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. Castaneda, M. Usuga-Castaneda, and J. Herrera-Ramirez, “Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs,” Appl. Opt. 46, 5321-5328 (2007).
[CrossRef] [PubMed]

2006

2005

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

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

2003

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

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

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

2000

1997

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

1995

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

1993

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

1992

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

1986

1981

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

1974

1968

1932

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227-1237 (1986).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

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 centered reduced moments. III: Spot moments-based method for laser characterization,” Opt. Commun. 248, 509-519 (2005).
[CrossRef]

Castaneda, R.

R. Castaneda, M. Usuga-Castaneda, and J. Herrera-Ramirez, “Experimental evidence of the spatial coherence moiré and the filtering of classes of radiator pairs,” Appl. Opt. 46, 5321-5328 (2007).
[CrossRef] [PubMed]

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, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[CrossRef]

R. Castaneda and J. Garcia-Sucerquia, “Electromagnetic spatial coherence wavelets,” J. Opt. Soc. Am. A 23, 81-90 (2006).
[CrossRef]

R. Castaneda, J. García, and J. Carrasquilla, “Quality descriptors of optical beams based on centered 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 and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226, 45-55 (2003).
[CrossRef]

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

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

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

Friberg, A. T.

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series, Vol. MS 69 (SPIE, 1993).

García, J.

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

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

Garcia-Sucerquia, J.

García-Sucerquia, J.

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

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

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

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]

Herrera-Ramirez, J.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

Mandel, L.

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

Marchand, E. W.

Mukunda, N.

Simon, R.

Usuga-Castaneda, M.

Walther, A.

Wigner, E. P.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Wolf, E.

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

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

E. W. Marchand and E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 641219-1226 (1974).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

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

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215-1224 (1981).
[CrossRef]

Opt. Commun.

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

R. Castaneda, “Tensor theory of electromagnetic radiometry,” Opt. Commun. 276, 14-30 (2007).
[CrossRef]

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

R. Castaneda, J. García, and J. Carrasquilla, “Quality descriptors of optical beams based on centered 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]

Phys. Rev.

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Other

A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, SPIE Milestone Series, Vol. MS 69 (SPIE, 1993).

D. Malacara, Optical Shop Testing, 2nd ed. (Wiley, 1992).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1993).

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

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Progress in Optics, E. Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 1-56.
[CrossRef]

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 (9)

Fig. 1
Fig. 1

Illustration of the center and difference coordinates for denoting pairs of points at both the AP plane and the OP plane.

Fig. 2
Fig. 2

Marginal power spectrum in Fraunhofer domain produced by a slit, of semiwidth a, under uniform illumination with a Schell model field (a) fully spatially coherent ( b = 0 ). (b) spatially partially coherent, and (c) spatially incoherent ( b a ).

Fig. 3
Fig. 3

Experimental setup for determining the map of classes of source pairs.

Fig. 4
Fig. 4

Experimental results obtained by using the setup in Fig. 3. Two different pinhole masks (left column) and three different support sizes for the Gaussian degree of spatial coherence were used. Images in the second and fourth rows are the corresponding maps of classes of source pairs to the interference patterns directly over each map.

Fig. 5
Fig. 5

Conceptual illustration of the support distortion by the aperture edge. The simple case of a circular shaped aperture and support is assumed.

Fig. 6
Fig. 6

Phase-space representation of Fraunhofer diffraction. Theoretical results were numerically calculated for diffraction through a slit of width 2 a . Experimental results were obtained by using the setup in Fig. 3 after attaching a circular aperture at the mask plane M.

Fig. 7
Fig. 7

Phase-space representation of Fresnel diffraction, assuming two Fresnel zones inscribed within the diffracting aperture. Theoretical results were numerically calculated for diffraction through a slit of width 2 a . Experimental results were obtained by using the setup in Fig. 3 after attaching a circular aperture at the mask plane M.

Fig. 8
Fig. 8

Transversal profiles of the power spectrum recorded by the CCD sensor of the experimental setup in Fig. 3, for different sizes of the aperture stop and values of the parameter E.

Fig. 9
Fig. 9

Area of the recorded power spectrum versus (a) the aperture diameter and (b) the natural logarithm of the parameter E values, by a complex degree of spatial coherence with support diameter of 1.5 mm .

Equations (18)

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

W ( r A + r D / 2 , r A r D / 2 , ξ A ; ω ) = S ( r A , ξ A , ω ) exp ( i k z r D ξ A ) ,
S ( r A , ξ A ; ω ) = W ( r A , r A , ξ A ; ω ) = AP μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) S ( ξ A + ξ D / 2 ; ω ) t ( ξ A + ξ D / 2 ) × S ( ξ A ξ D / 2 , ω ) t * ( ξ A ξ D / 2 ) exp ( i k z ξ A ξ D ) exp ( i k z ξ D r A ) d 2 ξ D
W ( r A + r D / 2 , r A r D / 2 ; ω ) = ( 1 λ z ) 2 exp ( i k z r A r D ) AP W ( r A + r D / 2 , r A r D / 2 , ξ A ; ω ) d 2 ξ A ,
S ( r A ; ω ) = ( 1 λ z ) 2 AP S ( r A , ξ A ; ω ) d 2 ξ A
OP S ( r A ; ω ) d 2 r A = AP S ( ξ A ; ω ) | t ( ξ A ) | 2 d 2 ξ A = ( 1 λ z ) 2 OP AP S ( r A , ξ A ; ω ) d 2 ξ A d 2 r A .
( 1 λ z ) 2 OP S ( r A , ξ A ; ω ) d 2 r A = S ( ξ A ; ω ) | t ( ξ A ) | 2
W ( r A + r D / 2 , r A r D / 2 ; ω ) = C ( 1 λ z ) 2 exp ( i k z r A r D ) AP S ( ξ A ; ω ) | t ( ξ A ) | 2 exp ( i k z r D ξ A ) d 2 ξ A ,
S ( r A , ξ A ; ω ) = AP μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) Ψ ( ξ A + ξ D / 2 ; ω ) Ψ * ( ξ A ξ D / 2 ; ω ) exp ( i k z ξ D r A ) d 2 ξ D .
S ( r A , ξ A ; ω ) = C S ( ξ A ; ω ) | t ( ξ A ) | 2 + 2 AP ξ D 0 | μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) | S ( ξ A + ξ D / 2 ; ω ) | t ( ξ A + ξ D / 2 ) | S ( ξ A ξ D / 2 , ω ) | t ( ξ A ξ D / 2 ) | cos [ k z ( ξ A r A ) ξ D + α 12 + ϕ 12 ] d 2 ξ D ,
OP cos [ k z ( ξ A r A ) · ξ D + α 12 + ϕ 12 ] d 2 r A = 0 ,
2 ( 1 λ z ) 2 exp ( i k z r A r D ) AP | μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) | S ( ξ A + ξ D / 2 ; ω ) | t ( ξ A + ξ D / 2 ) | × S ( ξ A ξ D / 2 , ω ) | t ( ξ A ξ D / 2 ) | cos [ k z r A ξ D α 12 ϕ 12 ] exp ( i k z r D ξ A ) d 2 ξ A
S ˜ ( ξ D ; ω ) = C ( 1 λ z ) 2 AP S ( ξ A ; ω ) | t ( ξ A ) | 2 d 2 ξ A δ ( ξ D ) + 2 AP ξ D 0 μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) × S ( ξ A + ξ D / 2 ; ω ) t ( ξ A + ξ D / 2 ) S ( ξ A ξ D / 2 , ω ) t * ( ξ A ξ D / 2 ) d 2 ξ A ,
S ˜ ( ξ A , ξ D ; ω ) = C S ( ξ A ; ω ) | t ( ξ A ) | 2 δ ( ξ D ) + 2 ( λ z ) 2 μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) × S ( ξ A + ξ D / 2 ; ω ) t ( ξ A + ξ D / 2 ) S ( ξ A ξ D / 2 , ω ) t * ( ξ A ξ D / 2 ) ,
S ( r A ; ω ) = ( 1 λ z ) 2 { C AP S ( ξ A ; ω ) | t ( ξ A ) | 2 d 2 ξ A + 2 AP AP ξ D 0 | μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) | S ( ξ A + ξ D / 2 ; ω ) × | t ( ξ A + ξ D / 2 ) | S ( ξ A ξ D / 2 ; ω ) | t ( ξ A ξ D / 2 ) | cos [ k z ( ξ A r A ) ξ D + α 12 + ϕ 12 ] d 2 ξ D d 2 ξ A } ,
S ( r A ; ω ) = S I ( r A ; ω ) + S C ( r A ; ω ) = ( 1 λ z ) 2 [ 0 | ξ A | b S I ( r A , ξ A ; ω ) d 2 ξ A + 0 | ξ A | b S C ( r A , ξ A ; ω ) d 2 ξ A ] ,
S I , C ( r A , ξ A ; ω ) = C S ( ξ A ; ω ) | t ( ξ A ) | 2 + 2 0 | ξ D | K | μ ( ξ A + ξ D / 2 , ξ A , ξ D / 2 ; ω ) | S ( ξ A + ξ D / 2 ; ω ) | t ( ξ A + ξ D / 2 ) | × S ( ξ A ξ D / 2 , ω ) | t ( ξ A ξ D / 2 ) | × cos [ k z ( ξ A r A ) ξ D + α 12 + ϕ 12 ] d 2 ξ D ,
S ( r A ; ω ) = ( 1 λ z ) 2 S o ( ω ) 0 < | ξ D | | ξ D | MAX 0 | ξ A | a μ ( ξ A + ξ D / 2 , ξ A ξ D / 2 ; ω ) d 2 ξ A exp ( i k z r A ξ D ) d 2 ξ D
μ ¯ ( ξ D ; ω ) = 1 A I S o ( ω ) OP S ( r A ; ω ) exp ( i k z r A ξ D ) d 2 r A .

Metrics