Abstract

We introduce a generalized angular spectrum representation for quantized light beams. By using our formalism, we are able to derive simple expressions for the electromagnetic vector potential operator in the case of (a) time-independent paraxial fields, (b) time-dependent paraxial fields, and (c) nonparaxial fields. For the first case the well-known paraxial results are fully recovered.

© 2006 Optical Society of America

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References

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  1. M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
    [CrossRef]
  2. N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
    [CrossRef]
  3. A. Aiello and J. P. Woerdman, arXiv.org e-Print archive, http://arxiv.org/quant-ph/0502164 (February 25, 2005).
  4. J. Visser, "Operator description of the dynamics of optical modes," Ph.D. dissertation (Leiden University, 2005).
  5. Very recently, the same dispersion relation we first introduced in Ref. has been used by G. F. Calvo, A. Picón, and E. Bagan, arXiv.org e-print archive, http://arxiv.org/quant-ph/0509040, to illustrate some interesting properties of photon angular momentum.
  6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).
  7. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).
  8. See Ref. for a full discussion about the physical meaning of the parameter thetav.
  9. I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
    [CrossRef] [PubMed]

2005 (2)

A. Aiello and J. P. Woerdman, arXiv.org e-Print archive, http://arxiv.org/quant-ph/0502164 (February 25, 2005).

J. Visser, "Operator description of the dynamics of optical modes," Ph.D. dissertation (Leiden University, 2005).

2002 (1)

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

2000 (1)

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

1999 (1)

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

1995 (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

1991 (1)

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

Aiello, A.

A. Aiello and J. P. Woerdman, arXiv.org e-Print archive, http://arxiv.org/quant-ph/0502164 (February 25, 2005).

Bagan, E.

Very recently, the same dispersion relation we first introduced in Ref. has been used by G. F. Calvo, A. Picón, and E. Bagan, arXiv.org e-print archive, http://arxiv.org/quant-ph/0509040, to illustrate some interesting properties of photon angular momentum.

Calvo, G. F.

Very recently, the same dispersion relation we first introduced in Ref. has been used by G. F. Calvo, A. Picón, and E. Bagan, arXiv.org e-print archive, http://arxiv.org/quant-ph/0509040, to illustrate some interesting properties of photon angular momentum.

Deutsch, I. H.

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

Garrison, J. C.

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

Gisin, N.

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

Kolobov, M. I.

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

Picón, A.

Very recently, the same dispersion relation we first introduced in Ref. has been used by G. F. Calvo, A. Picón, and E. Bagan, arXiv.org e-print archive, http://arxiv.org/quant-ph/0509040, to illustrate some interesting properties of photon angular momentum.

Ribody, G.

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

Visser, J.

J. Visser, "Operator description of the dynamics of optical modes," Ph.D. dissertation (Leiden University, 2005).

Woerdman, J. P.

A. Aiello and J. P. Woerdman, arXiv.org e-Print archive, http://arxiv.org/quant-ph/0502164 (February 25, 2005).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

Zbinden, H.

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

Phys. Rev. A (1)

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999).
[CrossRef]

N. Gisin, G. Ribody, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).
[CrossRef]

Other (6)

A. Aiello and J. P. Woerdman, arXiv.org e-Print archive, http://arxiv.org/quant-ph/0502164 (February 25, 2005).

J. Visser, "Operator description of the dynamics of optical modes," Ph.D. dissertation (Leiden University, 2005).

Very recently, the same dispersion relation we first introduced in Ref. has been used by G. F. Calvo, A. Picón, and E. Bagan, arXiv.org e-print archive, http://arxiv.org/quant-ph/0509040, to illustrate some interesting properties of photon angular momentum.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, 1995).

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford U. Press, 2000).

See Ref. for a full discussion about the physical meaning of the parameter thetav.

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Equations (24)

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A ̂ ( + ) ( r , t ) = d 3 k ( 16 π 3 ϵ 0 c k ) 1 2 × λ = 1 2 ϵ ( λ ) ( k ) a ̂ λ ( k ) exp ( i k r i c k t ) .
A ̂ ( + ) ( r , t ) = s = ± 1 d k x d k y 0 d ζ ( 16 π 3 ϵ 0 c k s ) 1 2 × λ = 1 2 ϵ ( λ ) ( k s ) a ̂ λ ( k s ) exp ( i k s r i c k s t ) ,
[ a ̂ λ ( k s ) , a ̂ λ ( k s ) ] = δ λ λ δ s s δ ( 2 ) ( q q ) δ ( ζ ζ ) .
k x = q x , k y = q y , ζ = f ( q , ω ) ,
I ω ( f , q ) = { ω R + : f ( q , ω ) 0 } .
δ ( 2 ) ( q q ) δ ( ζ ζ ) = δ ( 2 ) ( q q ) δ [ f ( q , ω ) f ( q , ω ) ] = δ ( 2 ) ( q q ) δ ( ω ω ) d f ( q , ω ) d ω .
[ a ̂ λ ( k s ) , a ̂ λ ( k s ) ] = δ λ λ δ s s δ ( 2 ) ( q q ) δ ( ω ω ) d f ( q , ω ) d ω .
a ̂ λ s ( q , ω ) a ̂ λ ( k s ) d f ( q , ω ) d ω .
[ a ̂ λ s ( q , ω ) , a ̂ λ s ( q , ω ) ] = δ λ λ δ s s δ ( 2 ) ( q q ) δ ( ω ω ) .
R 2 d 2 q I ω ( f , q ) d ω = R + d ω C q ( f , ω ) d 2 q .
A ̂ ( + ) ( r , t ) = s = ± 1 0 d ω C q ( f , ω ) d 2 q ( q 2 + f 2 16 π 3 ϵ 0 c ) 1 2 × λ = 1 2 ϵ ( λ ) ( q , s f ) a ̂ λ s ( q , ω ) d f d ω × exp ( i q x + i s f z i t c q 2 + f 2 ) ,
A ̂ ( + ) ( r , t ) = s = ± 1 0 d ω exp [ i ω ( t s z c ) ] Ψ ̂ s ( r , t ; ω ) ,
Ψ ̂ s ( r , t ; ω ) = C q ( f , ω ) d 2 q ( d f d ω 16 π 3 ϵ 0 c q 2 + f 2 ) 1 2 × λ = 1 2 ϵ s ( λ ) ( q , ω ) a ̂ λ s ( q , ω ) × exp [ i q x + i z s ( f ω c ) ] × exp [ i t ( c q 2 + f 2 ω ) ] .
ϵ s ( 1 ) ( q , ω ) = ( f q ̂ s q z ̂ ) q 2 + f 2 .
a ̂ λ ( k s ) a ̂ λ ( k s ) d ζ = a ̂ λ s ( q , ω ) a ̂ λ s ( q , ω ) d ω
H ̂ = 1 2 s = ± 1 0 d ω C q ( f , ω ) d 2 q c q 2 + f 2 ( q , ω ) × λ = 1 2 [ a ̂ λ s ( q , ω ) a ̂ λ s ( q , ω ) + a ̂ λ s ( q , ω ) a ̂ λ s ( q , ω ) ] .
2 Ψ ̂ s ( r ; ω ) x 2 + 2 Ψ ̂ s ( r ; ω ) y 2 + 2 i s ω c Ψ ̂ s ( r ; ω ) z = 0 .
f ( q , ω ) = ω c ( 1 q 2 c 2 2 ω 2 ) .
A ̂ ( + ) ( r , t ) = s = ± 1 0 d ω exp [ i ω ( t s z c ) ] × C q ( f , ω ) d 2 q [ ( 1 + ϑ 2 ) 16 π 3 ϵ 0 c ω 1 + ϑ 4 ] 1 2 × λ = 1 2 ϵ s ( λ ) ( q , ω ) a ̂ λ s ( q , ω ) exp ( i q x i s q 2 c 2 ω z ) × exp [ i ω t ( 1 + ϑ 4 1 ) ] .
2 Ψ ̂ s x 2 + 2 Ψ ̂ s y 2 + 2 i s ω c Ψ ̂ s z + 2 i ω c 2 Ψ ̂ s t = 0 ,
f ( q , ω ) = ω c ( 1 q 2 c 2 4 ω 2 ) .
A ̂ ( + ) ( r , t ) = s = ± 1 0 d ω exp [ i ω ( t s z c ) ] × C q ( f , ω ) d 2 q ( 16 π 3 ϵ 0 c ω ) 1 2 × λ = 1 2 ϵ s ( λ ) ( q , ω ) a ̂ λ s ( q , ω ) exp [ i q x i q 2 c 4 ω ( s z + c t ) ] .
f ( q , ω ) = ω c ( 1 q 2 c 2 ω 2 ) 1 2 ω c ( 1 q 2 c 2 2 ω 2 ) ,
A ̂ ( + ) ( r , t ) = s = ± 1 0 d ω exp [ i ω ( t s z c ) ] × C q ( f , ω ) d 2 q ( cos θ 16 π 3 ϵ 0 c ω ) 1 2 λ = 1 2 ϵ s ( λ ) ( q , ω ) a ̂ λ s ( q , ω ) × exp [ i q x i s z ω ( 1 cos θ ) c ] .

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