Abstract

The notion of a spherical angular spectrum leads to the decomposition of the field amplitude on a spherical emitter into a sum of spherical waves that converge onto the Fourier sphere of the emitter. Unlike the usual angular spectrum, the spherical angular spectrum is propagated as the field amplitude, in a way that can be expressed by a fractional order Fourier transform.

© 2006 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  2. P. Pellat-Finet and G. Bonnet, Opt. Commun. 111, 141 (1994).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  4. P. Pellat-Finet, Opt. Lett. 19, 1388 (1994).
    [CrossRef] [PubMed]
  5. V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
    [CrossRef]
  6. P. Pellat-Finet and E. Fogret, Opt. Commun. 258, 103 (2006).
    [CrossRef]
  7. P. Pellat-Finet and P. E. Durand, C. R. Phys. 7, 457 (2006).
    [CrossRef]

2006 (2)

P. Pellat-Finet and E. Fogret, Opt. Commun. 258, 103 (2006).
[CrossRef]

P. Pellat-Finet and P. E. Durand, C. R. Phys. 7, 457 (2006).
[CrossRef]

1994 (2)

P. Pellat-Finet, Opt. Lett. 19, 1388 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet and G. Bonnet, Opt. Commun. 111, 141 (1994).
[CrossRef]

1980 (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Bonnet, G.

P. Pellat-Finet and G. Bonnet, Opt. Commun. 111, 141 (1994).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

Durand, P. E.

P. Pellat-Finet and P. E. Durand, C. R. Phys. 7, 457 (2006).
[CrossRef]

Fogret, E.

P. Pellat-Finet and E. Fogret, Opt. Commun. 258, 103 (2006).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Namias, V.

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet and E. Fogret, Opt. Commun. 258, 103 (2006).
[CrossRef]

P. Pellat-Finet and P. E. Durand, C. R. Phys. 7, 457 (2006).
[CrossRef]

P. Pellat-Finet, Opt. Lett. 19, 1388 (1994).
[CrossRef] [PubMed]

P. Pellat-Finet and G. Bonnet, Opt. Commun. 111, 141 (1994).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

C. R. Phys. (1)

P. Pellat-Finet and P. E. Durand, C. R. Phys. 7, 457 (2006).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, J. Inst. Math. Appl. 25, 241 (1980).
[CrossRef]

Opt. Commun. (2)

P. Pellat-Finet and E. Fogret, Opt. Commun. 258, 103 (2006).
[CrossRef]

P. Pellat-Finet and G. Bonnet, Opt. Commun. 111, 141 (1994).
[CrossRef]

Opt. Lett. (1)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

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Figures (2)

Fig. 1
Fig. 1

Spherical wave converging at the point R A Φ on the Fourier sphere F generates a field amplitude on A equal to exp ( 2 i π Φ r λ ) .

Fig. 2
Fig. 2

Field transfer from A to B can be decomposed into a linear filtering from A to A followed by a Fourier transform from A to B and another linear filtering from B to B .

Equations (27)

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U Q ( s ) = R 2 h ( s r ) U P ( r ) d r ,
h ( r ) = ( i λ D ) exp [ ( i π λ D ) r 2 ] ,
A P ( Φ ) = 1 λ 2 F π 2 [ U P ] ( Φ λ ) = 1 λ 2 R 2 exp ( 2 i π λ Φ r ) U P ( r ) d r ,
H ( Φ ) = F π 2 [ h ] ( Φ λ ) = exp [ ( i π D λ ) Φ 2 ] ,
A Q ( Φ ) = H ( Φ ) A P ( Φ ) .
S A ( Φ ) = ( 1 λ 2 ) F π 2 [ U A ] ( Φ λ ) .
U A ( r ) = R 2 exp ( 2 i π λ Φ r ) S A ( Φ ) d Φ ,
U B ( s ) = i λ D exp [ i π s 2 λ ( 1 D + 1 R B ) ] R 2 exp ( 2 i π λ D s r ) × exp [ i π r 2 λ ( 1 R A 1 D ) ] U A ( r ) d r .
U F ( s ) = ( i λ R A ) F π 2 [ U A ] ( s λ R A ) ,
S F ( Φ ) = 1 λ 2 F π 2 [ U F ] ( Φ λ ) = i R A λ F π 2 [ S A ] ( R A Φ λ ) .
h B A ( r ) = i λ D exp [ ( i π κ λ D ) r 2 ] .
U B ( s ) = R 2 h B A ( s κ r ) U A ( r ) d r = [ h B A U A ] ( s κ )
S B ( Φ ) = κ exp [ ( i π κ D λ ) Φ 2 ] S A ( κ Φ ) .
S B ( Φ ) = i R A κ λ κ exp ( i π κ D Φ 2 λ ) R 2 exp ( i π D Φ 2 λ κ ) × exp ( 2 i π κ R A λ κ Φ Φ ) S A ( Φ ) d Φ .
F α [ f ] ( ρ ) = i e i α sin α exp ( i π ρ 2 cot α ) R 2 exp ( i π ρ 2 cot α ) × exp ( 2 i π ρ ρ sin α ) f ( ρ ) d ρ .
cot 2 α = ( D + R B ) ( R A D ) D ( D R A + R B ) , α D 0 ,
ϵ 2 = D ( D + R B ) ( R A D ) ( D R A + R B ) , ϵ R A > 0 .
ρ = r ( λ ϵ R A ) 1 2 , V A ( ρ ) = U A [ ( λ ϵ R A ) 1 2 ρ ] ,
σ = ( cos α + ϵ sin α ) s ( λ ϵ R A ) 1 2 , V B ( σ ) = U B [ ( λ ϵ R A ) 1 2 σ cos α + ϵ sin α ] .
V B ( σ ) = e i α ( cos α + ϵ sin α ) F α [ V A ] ( σ )
ϕ = ( ϵ R A λ ) 1 2 Φ , ϕ = ( ϵ R A λ ) 1 2 Φ cos α + ϵ sin α ,
Σ A ( ϕ ) = S A [ ( λ ϵ R A ) 1 2 ϕ ] ,
Σ B ( ϕ ) = S B [ ( λ ϵ R A ) 1 2 ( cos α + ϵ sin α ) ϕ ] .
Σ B ( ϕ ) = [ e i α ( cos α + ϵ sin α ) ] F α [ Σ A ] ( ϕ ) .
Σ A ( ϕ ) = ( ϵ R A λ ) V ̂ A ( ϕ ) ,
Σ B ( ϕ ) = { ϵ R A [ λ ( cos α + ϵ sin α ) 2 ] } V ̂ B ( ϕ ) ,
V ̂ B ( ϕ ) = e i α ( cos α + ϵ sin α ) F α [ V ̂ A ] ( ϕ ) ,

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