Abstract

Due to analytical and numerical difficulties, the propagation of optical fields in any state of spatial coherence is traditionally computed under severe approximations. The paraxial approach in the Fresnel–Fraunhofer domain is one of the most widely used. These approximations provide a rough knowledge of the actual light behavior as it propagates, which is not enough for supporting applications, such as light propagation under a high numerical aperture (NA). In this paper, a non-approximated model for the propagation of optical fields in any state of spatial coherence is presented. The method is applicable in very practical cases, as high-NA propagations, because of its simplicity of implementation. This approach allows for studying unaware behaviors of light as it propagates. The light behavior close to the diffracting transmittances can also be analyzed with the aid of the proposed tool.

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References

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  1. F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica 5(8), 785–795 (1938).
    [Crossref]
  2. M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon Press, 1993), Chap. 10.
  3. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Chap. 4.
  4. Z. Jaroszewicz, Axicons: Design and Propagation Properties (Research and development treatises, SPIE Polish chapter) (SPIE, 1997), Vol. 5.
  5. D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci. 349(2), 637–640 (2010).
    [Crossref] [PubMed]
  6. K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. 283(24), 4859–4865 (2010).
    [Crossref]
  7. R. Castañeda, “The optics of spatial coherence wavelets, in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic Press, 2010), Vol. 164.
  8. R. Castañeda, G. Cañas-Cardona, and J. Garcia-Sucerquia, “Radiant, virtual, and dual sources of optical fields in any state of spatial coherence,” J. Opt. Soc. Am. A 27(6), 1322–1330 (2010).
    [Crossref] [PubMed]
  9. R. Castañeda, H. Muñoz-Ossa, and J. Garcia-Sucerquia, “Efficient numerical calculation of interference and diffraction of optical fields in any state of spatial coherence in the phase-space representation,” Appl. Opt. 49(31), 6063–6071 (2010).
    [Crossref]
  10. R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. ,  58(11) (2011), doi: .
    [Crossref]
  11. K. Iizuka, Engineering Optics (Springer Verlag, 1985).
  12. R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226(1-6), 45–55 (2003).
    [Crossref]

2011 (1)

R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. ,  58(11) (2011), doi: .
[Crossref]

2010 (4)

2003 (1)

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226(1-6), 45–55 (2003).
[Crossref]

1938 (1)

F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica 5(8), 785–795 (1938).
[Crossref]

Alvarez-Palacio, D. C.

D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci. 349(2), 637–640 (2010).
[Crossref] [PubMed]

Bokor, N.

K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. 283(24), 4859–4865 (2010).
[Crossref]

Cañas-Cardona, G.

Castañeda, R.

García, J.

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226(1-6), 45–55 (2003).
[Crossref]

Garcia-Sucerquia, J.

Jahn, K.

K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. 283(24), 4859–4865 (2010).
[Crossref]

Muñoz, H.

R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. ,  58(11) (2011), doi: .
[Crossref]

Muñoz-Ossa, H.

Zernike, F.

F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica 5(8), 785–795 (1938).
[Crossref]

Appl. Opt. (1)

J. Colloid Interface Sci. (1)

D. C. Alvarez-Palacio and J. Garcia-Sucerquia, “Lensless microscopy technique for static and dynamic colloidal systems,” J. Colloid Interface Sci. 349(2), 637–640 (2010).
[Crossref] [PubMed]

J. Mod. Opt. (1)

R. Castañeda, H. Muñoz, and G. Cañas-Cardona, “The structured spatial coherence support,” J. Mod. Opt. ,  58(11) (2011), doi: .
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

R. Castañeda and J. García, “Classes of source pairs in interference and diffraction,” Opt. Commun. 226(1-6), 45–55 (2003).
[Crossref]

K. Jahn and N. Bokor, “Intensity control of the focal spot by vectorial beam shaping,” Opt. Commun. 283(24), 4859–4865 (2010).
[Crossref]

Physica (1)

F. Zernike, “The concept of Degree of Coherence and its application to optical problems,” Physica 5(8), 785–795 (1938).
[Crossref]

Other (5)

M. Born and E. Wolf, Principles of Optics, 6th. ed. (Pergamon Press, 1993), Chap. 10.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995), Chap. 4.

Z. Jaroszewicz, Axicons: Design and Propagation Properties (Research and development treatises, SPIE Polish chapter) (SPIE, 1997), Vol. 5.

R. Castañeda, “The optics of spatial coherence wavelets, in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Academic Press, 2010), Vol. 164.

K. Iizuka, Engineering Optics (Springer Verlag, 1985).

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Figures (8)

Fig. 1
Fig. 1

Diffraction geometry related to the structured spatial coherence support centered at the point ξ A on the AP, which determines the marginal power spectrum S( ξ A , r A ;ν ) . The size of d 2 ξ D grows from null (the point ξ A ) until it covers the integration area.

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Fig. 2
Fig. 2

Comparing (a) the exact calculation with (b) the paraxial approach for the light emitted by a single radiant point source, placed at ξ A =0.5μ m on the AP, in the Fresnel–Fraunhofer domain. The exact calculation predicts the Lorentzian-like free-space diffraction envelope. The power spectrum profiles (bottom row) were scaled for presentation purposes.

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Fig. 3
Fig. 3

(a) Profiles of the power spectrum at the OP for different distances z. (b) Comparison of the free-space diffraction envelope in the Fresnel–Fraunhofer domain with conventional Lonrentzian and Gaussian distributions (the Lorentzian-like profile of the free-space diffraction envelope is apparent). (c) Encircled energy curves of the power spectrum at the OP for different distances z; 95% of the total energy spreads within the NA = 0.97. All the profiles were scaled for presentation purposes.

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Fig. 4
Fig. 4

(a) Exactly calculated and (b) paraxial approached marginal power spectra of a fully spatially coherent Young’s experiment. (c) Comparison between the exactly calculated and the paraxial approached power spectra profiles in the paraxial region of the Fresnel–Fraunhofer domain for | b |=0.1μ m (top row), | b |=3μ m (middle row), and | b |=10μ m (bottom row).

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Fig. 5
Fig. 5

(a) Power spectrum calculated exactly propagation along the z axis in a Young’s experiment with spatially partially coherent point sources. (b) Exactly calculated power spectrum profile at the OP in the Fresnel–Fraunhofer domain for the Young’s experiment in (a). (c) Comparison between the exactly calculated and the paraxial approached power spectra fringe patterns in the Fresnel–Fraunhofer domain. Fully spatially coherent point sources with 50μ m of separation are regarded for modeling the Young’s experiment, and the paraxial region of the fringe pattern is considered.

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Fig. 6
Fig. 6

Exact calculation of interference and diffraction produced by an array of seven fully spatially coherent radiant point sources in the Fresnel–Fraunhofer domain for λ=0.632μ m and different pitches and array lengths. (a) Marginal power spectra, (b) propagation of the power spectra in the first 10μ m along the z axis from the AP, and (c) power spectra profiles at the OP in the Fresnel–Fraunhofer domain also indicating the pitch and total length of the array.

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Fig. 7
Fig. 7

Comparison between the Fraunhofer power spectra obtained by the exact calculation with a discrete set of fully spatially coherent radiant point sources, of pitch |  b | and array length L, and from the paraxial diffraction of a uniform and spatially coherent plane wave by a slit whose width equals the array length for (a) N=7 radiant point sources and (b) N=16 radiant point sources. (c) Effect of reducing the total length of the source array ( N=7 ) onto the power spectrum at the OP.

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Fig. 8
Fig. 8

Modulating energy at the AP in a Young’s experiment with fully spatially coherent radiant point sources. (a) Marginal power spectrum on the top and power spectrum profile on the bottom. (b) Modulating energy profiles for phase difference of 2nπ , with n=0, 1, ⋯ , between the emissions of the sources on the top, and of ( 2n+1 )π on the bottom. (c) Modulating energy profiles for phase difference of ( 2n+1/2 )π on the top and of ( 2n+3/2 )π on the bottom.

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Tables (1)

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Table 1 Values for the Indices P and Q of the Summations in Eqs. (11) and (12)for 15 Classes of Radiator Pairs

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Equations (12)

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W( P 1 , P 2 ;ν )= ∫ AP ∫ AP W( Q 1 , Q 2 ;ν )  exp[  i k ( s 1 − s 2 ) ] s 1   s 2  Λ( ϑ 1 )  Λ * ( ϑ 2 ) d σ 1  d σ 2 ,
S( r A ;ν )=W( r A , r A ;ν )= ∫ AP S( ξ A , r A ;ν )  d 2 ξ A ,
S( ξ A , r A ;ν )= 1 4 λ 2   ∫ AP S 0 ( ξ A + ξ D /2 )  t( ξ A + ξ D /2 )  S 0 ( ξ A − ξ D /2 )   t * ( ξ A − ξ D /2 )  × μ( ξ A + ξ D /2  ,  ξ A − ξ D /2 ) exp[ i k  z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 + ξ A ⋅ ξ D − r A ⋅ ξ D ] z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 + ξ A ⋅ ξ D − r A ⋅ ξ D   ×   exp[ −i k  z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 − ξ A ⋅ ξ D + r A ⋅ ξ D ] z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 − ξ A ⋅ ξ D + r A ⋅ ξ D   ( z+ z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 + ξ A ⋅ ξ D − r A ⋅ ξ D )( z+ z 2 + | r A − ξ A | 2 + | ξ D | 2 /4 − ξ A ⋅ ξ D + r A ⋅ ξ D )  d 2 ξ D .
S( ξ A , r A ;ν )= S 0 ( ξ A ) 4 λ 2   ( z+ z 2 + |   r A − ξ A | 2 z 2 + |   r A − ξ A | 2 ) 2  δ( ξ A −a )
S( r A ;ν )= S 0 ( a ) 4 λ 2   ( z+ z 2 + |   r A −a | 2 z 2 + |   r A −a | 2 ) 2 ,
S( r A ;ν )= ( 1 λ z ) 2  {   S 0 ( a+b/2 ) +  S 0 ( a−b/2 ) + 2  S 0 ( a+b/2 )   S 0 ( a−b/2 )      × | μ( a+b/2  , a−b/2 )  | cos[ k z  ( r A −a )⋅b−α( a+b/2  , a−b/2 ) ]  }.
S( ξ A , r A ;ν )= S 0 ( ξ A ) 4 λ  2   ( z+ z 2 + | r A − ξ A | 2 z 2 + | r A − ξ A | 2 ) 2  [ δ( ξ A −a+b/2 ) + δ( ξ A −a−b/2 ) ] + 1 2 λ  2  δ( ξ A −a )  S 0 ( ξ A +b/2 )   S 0 ( ξ A −b/2 )  | μ( ξ A +b/2  ,  ξ A −b/2 )  | × ( z+ z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b )  ( z+ z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b ) × cos[ k  z 2 + | r A − ξ A | 2 + | b | 2 /4 + ξ A ⋅b− r A ⋅b  − k  z 2 + | r A − ξ A | 2 + | b | 2 /4 − ξ A ⋅b+ r A ⋅b  + α( ξ A +b/2  ,  ξ A −b/2 ) ]
S( r A ;ν )= S 0 ( a−b/2 ) 4 λ 2   ( z+ z 2 + | r A −a+b/2 | 2 z 2 + | r A −a+b/2 | 2 ) 2  +  S 0 ( a+b/2 ) 4 λ 2   ( z+ z 2 + | r A −a−b/2 | 2 z 2 + | r A −a−b/2 | 2 ) 2 + 1 2 λ 2   S 0 ( a+b/2 )   S 0 ( a−b/2 )  | μ( a+b/2  , a−b/2 )  | × ( z+ z 2 + | r A −a | 2 + | b | 2 /4 +a⋅b− r A ⋅b z 2 + | r A −a | 2 + | b | 2 /4 +a⋅b− r A ⋅b )  ( z+ z 2 + | r A −a | 2 + | b | 2 /4 −a⋅b+ r A ⋅b z 2 + | r A −a | 2 + | b | 2 /4 −a⋅b+ r A ⋅b ) × cos[ k  z 2 + | r A −a | 2 + | b | 2 /4 +a⋅b− r A ⋅b  − k  z 2 + | r A −a | 2 + | b | 2 /4 −a⋅b+ r A ⋅b  + α( a+b/2  , a−b/2 ) ],
S rad ( ξ A , r A ;ν )= S 0 ( ξ A ) 4 λ 2   ( z+ z 2 + |   r A − ξ A | 2 z 2 + |   r A − ξ A | 2 ) 2   ∑ n=0 N−1 δ( ξ A −n b )
S rad ( r A ;ν )= ∑ n=0 N−1 S 0 ( n b ) 4 λ 2   ( z+ z 2 + |   r A −n b | 2 z 2 + |   r A −n b | 2 ) 2 .
S virt ( m ) ( ξ A , r A ;ν )= 1 2 λ 2  [ ∑ n=P Q δ( ξ A −( n+β ) b ) ]  S 0 ( ξ A + m b /2 )   S 0 ( ξ A −m b/2 )  | μ( ξ A + m b /2  ,  ξ A −m b/2 )  | × ( z+ z 2 + | r A − ξ A | 2 + | m b | 2 /4 + ξ A ⋅m b− r A ⋅m b z 2 + | r A − ξ A | 2 + | m b | 2 /4 + ξ A ⋅m b− r A ⋅m b )  ( z+ z 2 + | r A − ξ A | 2 + | m b | 2 /4 − ξ A ⋅m b+ r A ⋅m b z 2 + | r A − ξ A | 2 + | m b | 2 /4 − ξ A ⋅m b+ r A ⋅m b ) × cos[ k  z 2 + | r A − ξ A | 2 + | m b | 2 /4 + ξ A ⋅m b− r A ⋅m b  − k  z 2 + | r A − ξ A | 2 + | m b | 2 /4 − ξ A ⋅m b+ r A ⋅m b  + α( ξ A +m b/2  ,  ξ A −m b/2 ) ]
S virt ( m ) ( r A ;ν )= 1 2 λ 2   ∑ n=P Q S 0 ( ( n+β+m/2 ) b )   S 0 ( ( n+β−m/2 ) b )  | μ( ( n+β+m/2 ) b , ( n+β−m/2 ) b )  | × ( z+ z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 +( n+β ) m ]  | b | 2 − r A ⋅m b z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 +( n+β ) m ]  | b | 2 − r A ⋅m b )  ( z+ z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 −( n+β ) m ]  | b | 2 + r A ⋅m b z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 −( n+β ) m ]  | b | 2 + r A ⋅m b ) × cos [ k  z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 +( n+β ) m ]  | b | 2 − r A ⋅m b  − k  z 2 + | r A −( n+β ) b | 2 +[ m 2 /4 −( n+β ) m ]  | b | 2 + r A ⋅m b  + α( ( n+β+m/2 ) b , ( n+β−m/2 ) b ) ].

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